Issue 
A&A
Volume 568, August 2014



Article Number  A57  
Number of page(s)  14  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201423761  
Published online  14 August 2014 
Modeling the cross power spectrum of the SunyaevZel’dovich and Xray surveys
Institut d’Astrophysique Spatiale, CNRS (UMR 8617) and Université ParisSud
11,
Bâtiment 121,
91405
Orsay,
France
email:
ghurier@ias.upsud.fr
Received:
5
March
2014
Accepted:
25
May
2014
The thermal SunyaevZel’dovich (tSZ) effect and Xray emission from galaxy clusters have been used extensively to constrain cosmological parameters. These constraints are highly sensitive to the relations between cluster masses and observables (tSZ and Xray fluxes). The crosscorrelation of tSZ and Xray data is thus a powerful tool, in addition of tSZ and Xray based analysis, for testing our modeling of both tSZ and Xray emission from galaxy clusters. We chose to explore this crosscorrelation because both emissions trace the hot gas in galaxy clusters and thus constitute one of the easiest correlations that can be studied. We present a complete modeling of the crosscorrelation between tSZ effect and Xray emission from galaxy clusters and focus on the dependencies with cluster scaling laws and cosmological parameters. We show that current knowledge of cosmological parameters and scalinglaw parameters leads to uncertainties of 48% on the overall normalization of the tSZX crosscorrelation power spectrum. We present the expected signaltonoise ratio for the tSZX crosscorrelation angular power spectrum and consider the sensitivity of actual tSZ and Xray surveys from Plancklike data and ROSAT. We demonstrate that this signaltonoise can reach 31.5 in a realistic situation, leading to a constraint on the amplitude of tSZX crosscorrelation up to 3.2%, which is ten times better than actual modeling limitations. Consequently, using it in addition to other probes of cosmological parameters and scaling relations, we show that the tSZX is a powerful probe that constrains the cosmological parameters of scaling relations.
Key words: galaxies: clusters: general / Xrays: galaxies: clusters / submillimeter: general / largescale structure of Universe / galaxies: clusters: intracluster medium
© ESO, 2014
1. Introduction
Galaxy clusters are the largest virialized structures. They can be observed through Xray, via the bremsstrahlung emission produced by the ionized intracluster medium (see, e.g., Bohringer et al. 2000; Ebeling et al. 2000, 2001). This hot intracluster medium also produces a distortion of the cosmic microwave background (CMB) blackbody emission via the thermal SunyaevZel’dovich (tSZ) effect (Sunyaev & Zeldovich 1969, 1972). This effect was observed toward a large number of clusters by Planck (Planck Collaboration VIII 2011; Planck Collaboration XXIX 2014), ACT (Marriage et al. 2011), and SPT (Reichardt et al. 2013).
The number of galaxy clusters is extremely sensitive to cosmological parameters, especially to the normalization of the matter power spectrum, σ_{8}, and to the matter density, Ω_{m}. It is thus possible to use galaxy cluster catalogs to constraint cosmological parameters (Vanderlinde et al. 2010; Sehgal et al. 2011; Planck Collaboration XX 2014) through a halo massfunction formalism.
We now have access to a full sky coverage for both Xray emission with the ROSAT all sky survey (RASS) and tSZ emission with Planck (Planck Collaboration VIII 2011). Consequently, beyond tSZ clusters catalogs analysis, it is possible to perform tSZ angular power spectrum analysis. This process allows to consider all clusters on the covered sky without any selection function (see, e.g., Planck Collaboration VIII 2011), contrary to catalogbased analysis. This allows to catch the signals from higher redshift and lower mass objects that are not detected individually. Such measurement is limited by the contamination produced by other astrophysical components, mainly the cosmic infrared background (CIB, Puget et al. 1996; Fixsen et al. 1998).
It is difficult to perform the same power spectrum analysis with Xray surveys. The Xray photons, at low energy (<0.5 keV), are absorbed by neutral hydrogen of our Galaxy and, at higher energy, the Xray sky power is dominated by the emission from active galactic nuclei (AGN). Consequently, Xray surveys are most commonly used to constraint the AGN spatial clustering (Krumpe et al. 2010; Miyaji et al. 2011; Krumpe et al. 2012).
In addition to autocorrelation power spectrum analysis, the cross power spectrum between tSZ effect and Xray emission can be used. This crosscorrelation is one of the easiest correlations to study, since both signals are produced by the same hot gas of electrons. Using such an approach allows the contamination to be minimized by other astrophysical components and suppresses the instrumental noise contribution to the power spectrum.
The tSZX cross spectrum is sensitive to both Xray and tSZ scaling relations (see, e.g, Benson et al. 2013; Planck Collaboration XI 2011; Arnaud et al. 2010, for present constraints on scaling relations). This sensitivity limits the determination of cosmological parameters. However, it offers another possibility for constraining tSZ and Xray scaling laws.
The utilization of the tSZX correlation has already been discussed in the literature. Diego et al. (2003) has attempted to directly compare theoretical predictions with the measured cross power spectrum between WMAP temperature anisotropy maps and ROSAT All Sky Survey (RASS). The limited sensitivity and resolution of the WMAP experiment leads to upper limits on the tSZX correlation. More recently, Hajian et al. (2013) have performed the measurement of the crosscorrelation between the tSZ sky and an Xray based catalog of clusters. From their analysis they derive σ_{8}(Ω_{m}/ 0.30)^{0.26} = 0.80 ± 0.02.
We present an uptodate modeling of the tSZ/Xray crosscorrelation. In Sect. 2, we present our modeling of the tSZX crosscorrelation. We give particular attention to the distribution in mass and redshift of the tSZX power. In Sect. 3, we explore the variations in the tSZX spectrum with respect to cosmological and scalinglaw parameters. We also discuss modeling uncertainties considering our knowledge on cosmological and scaling law parameters and present the main limitations for the tSZX correlation measurement using simulated Plancklike data. Finally in Sect. 4, we predict the expected signaltonoise for the tSZX correlation from simulations of Plancklike tSZ survey and ROSAT All Sky Survey, and we present the associated constraints on cosmological, scalinglaw, and profile parameters.
Throughout the paper, we used the PlanckCMB bestfitting cosmology (Planck Collaboration XVI 2014) as our fiducial cosmological model, unless otherwise specified. Thus, we consider H_{0} = 67.1 ± 1.4 km/s/Mpc, σ_{8} = 0.834 ± 0.027, and Ω_{m} = 0.317 ± 0.020.
2. Modeling tSZXray crosscorrelation
2.1. The tSZ effect from galaxy clusters
The tSZ effect consists of a small spectral distortion of the CMB black body (Sunyaev & Zeldovich 1969, 1972), and its intensity is related to the integral of the pressure across the line of sight via the Compton parameter. This parameter in a given direction of the sky reads as (1)where dl is the distance along the lineofsight, and n_{e} and T_{e} are the electron number density and the temperature, respectively.
In units of CMB temperature, the contribution of the tSZ effect to the submillimeter sky intensity for a given observation frequency ν is given by . Neglecting relativistic corrections, we have , with x = hν/ (k_{B}T_{CMB}). This function is equal to 0 around 217 GHz and is negative at lower frequencies and positive for higher frequencies. Thus the spectral distortion induced by the hot gas of baryons provides a characteristic signal allowing the pressure distribution to be directly measured in galaxy clusters. In the context of a ΛCDM cosmology, this spectral distortion is known to be independent of the redshift. This has been tested and validated for a redshift range from 0 to 1 (Hurier et al. 2014). This characteristic spectral distortion can be used to separate the tSZ from other emissions of the microwave sky to derive Compton parameter map (see, e.g, Remazeilles et al. 2011; Hurier et al. 2013; Planck Collaboration VIII 2011).
2.2. The Xray emission from galaxy clusters
The ionized gas in the intracluster medium produces an Xray emission via Bremsstrahlung. This radiation is proportional to the square of the electronic density. The energy spectrum of the Xray emission from galaxy clusters depends mainly on the temperature, T_{500}, of the intracluster medium and„ to a lesser extent on the metallicity, Z, of the gas. To model the metallicity evolution history, we follow Andreon (2012) using the relation derived from the analysis of 130 galaxy clusters in a redshift range from 0.1 to 1.3, (2)with t(z) the age of the universe at a redshift z.
From an observational point of view, the Xray spectrum depends on the redshift, z. Xray photons are also absorbed by the neutral hydrogen in our Galaxy. This absorption is particularly significant for photons with E < 0.5 keV. Consequently, the observed count rate depends on the column density of neutral hydrogen, n_{H}, on the line of sight.
To estimate the Xray flux from each cluster, we computed an unabsorbed Xray spectrum, φ_{unabs} = dn_{γ}/ dE, with n_{γ} the number of emitted photons by the cluster inside a radius of R_{500}, using a MEKAL model (Mewe et al. 1985). To do this, we used a relation between the physical properties of the cluster, mass, and redshift and the temperature. These relations are presented in Sect. 2.5. Then, we computed the absorbed spectrum,φ_{abs}, as (3)with σ(E) the photoelectric cross section. The unabsorbed luminosity, L_{500}, in a given energy bin [ E_{min},E_{max} ] of a cluster reads as(4)Finally, the expected number count in a given energy bin is computed as (5)where χ(z) is the comoving angular distance at redshift z, a the scale factor, A(E) the effective area of the detector as a function of the energy, ℳ(E′,E) is the energy redistribution matrix, E and E′ are the photon energy^{1} and the measured value of photon energy, respectively.
We define the flux to countrate conversion factor as CR(z,Z,n_{H},T_{500}) = S_{500}/L_{500}. However, clusters can be located at any position on the sky, so we have to convolve the CR factor by the distribution of n_{H} on the sky: (6)where is the probability to have a column density of hydrogen n_{H} on the line of sight.
2.3. The tSZXray cross power spectra
Decomposing both the tSZ Compton parameter map and Xray countrate map, we define Thus, the power spectra of both tSZ effect and Xray can be written as and the angular cross power spectrum of tSZ effect and Xray countrate map reads as(11)To model this crosscorrelation or the auto correlation power spectra, we assume the following general expression (12)where is the Poissonian contribution, is the twohalo term that accounts for correlation in the spatial distribution of clusters over the sky, and is produced by the warmhot intergalactic medium (WHIM). In the following, considering the low density and the low temperature of the WHIM, we assume , and thus we neglect his contribution to the total power spectrum.
The Poissonian term can be computed by assuming the Fourier transform of tSZ and Xray projected profiles weighted by the mass function, presented in Sect. 2.4, and the fluxes for tSZ effect and Xray count rate (see, e.g, Komatsu & Seljak 2002, for a derivation of the tSZ angular power spectrum): (13)
Scalinglaw parameters and error budget for both Y_{500} − M_{500} (Planck Collaboration XX 2014), L_{500} − M_{500} (Arnaud et al. 2010), and Y_{500} − T_{500} (Planck Collaboration XX 2014) relations.
where , the flux to countrate conversion factor described in Sect. 2.2, the clusters mass function described in Sect. 2.4, and the element of comoving volume. The term (1 + ρ_{YL}σ_{log Y}σ_{log L}) accounts for extra power produced by the scatter in the scaling relations described at Sect. 2.5.
The Fourier transform of the 3D profile projected across the line of sight on the sphere reads as (14)where p(r_{s}) is either the tSZ 3D profile or the Xray countrate 3D profile, and r_{s} = r/r_{o}, ℓ_{s} = D_{A}(z) /r_{o}, r_{o} is the scale radius of the profile.
The contribution of the twohalo term corresponds to large scale fluctuations in the matter power spectrum, which induce correlation in the cluster distribution over the sky. It can be computed as (see, e.g, Taburet et al. 2011, and references therein) (15)with P(k,z) the matter power spectrum computed using CLASS (Lesgourgues 2011), and b_{lin}(M,z) the time dependent linear bias factor that relates P(k,z) to the power spectrum of the cluster distribution over the sky. Following Mo & White (1996) and Komatsu & Kitayama (1999), we adopt b_{lin}(M,z) = 1 + (ν^{2}(M,z) − 1) /δ_{c}(z), with , D_{g}(z) is the linear growth factor, and δ_{c}(z) is the overdensity threshold for spherical collapse.
2.4. Mass function
Our computation of the tSZX correlation assumes the mass function calibrated on a numerical simulation from Tinker et al. (2008): (16)with (17)and ρ_{m}(z = 0) the mean matter density today. The coefficients A_{0}, A_{1}, A_{2}, and A_{3} are given in Tinker et al. (2008) for various overdensities, Δ_{mean}, with respect to the redshiftdependent mean cosmic density. These coefficients are interpolated to match Δ_{critical} defined with respect to the critical density. The relation between Δ_{critical} and Δ_{mean} is given by Δ_{mean} = Δ_{critical}/ Ω_{m}(z), with Ω_{m}(z) the matter density parameter at redshift z. The standard deviation of the density perturbation in a sphere or radius R, σ, is computed in linear perturbation theory.
2.5. tSZ and Xray fluxes
A key step in the modeling of the crosscorrelation between tSZ and Xray is to relate the mass, M_{500}, and the redshift, z, of a given cluster to tSZ flux, Y_{500}, and Xray luminosity L_{500} in the [E_{min} − E_{max}] keV energy band^{2} in the rest frame of the cluster. The crosscorrelation between tSZ effect and Xray emission is thus highly dependent on the M_{500} − Y_{500} and the M_{500} − L_{500} relations in terms of normalization and slope. Consequently, we need to use the relations derived from a representative sample of galaxy clusters, with careful propagation of statistical and systematic uncertainties. We stress that for power spectrum analysis, the intrinsic scatter of such scaling laws has to be considered, because it will produce extra power that has to be accounted for in order to avoid biases.
We used the M_{500} − Y_{500} scaling laws presented in Planck Collaboration XX (2014), (18)with E(z) = Ω_{m}(1 + z)^{3} + Ω_{Λ}. The coefficients Y_{⋆}, α_{sz}, and β_{sz} from Planck Collaboration XX (2014), are given in Table 1. We used b = 0.2 for the bias between Xray estimated mass and effective mass of the clusters. To model the L_{500} − M_{500} relation, we used the relation derived by Arnaud et al. (2010) from the REXCESS sample (Böhringer et al. 2007): (19)where M_{x,500} is the cluster mass estimated from Xray observations. It is related to the true mass M_{500} through M_{x,500} = (1 − b)M_{500}. The coefficients L_{⋆}, α_{x}, and β_{x} are given in Table 1.
The evolution of the M_{500} − L_{500} relation suffers from large uncertainties (Reichert et al. 2011). Consequently, we fixed β_{x} to a selfsimilar value of 7/3. However, we verified that variations from 2 to 3 do not change our results with respect to other sources of uncertainties on the M_{500} − L_{500} relation. Indeed, Xray signal is dominated by low redshift clusters, and thus presents small variations with the redshift evolution of the M_{500} − L_{500} relation.
The two relations, M_{500} − Y_{500} and M_{500} − L_{500}, have intrinsic scatters, σ_{log Y} = 0.075 and σ_{log L} = 0.183, respectively. These scatters will contribute to the total power measured on the sky. Indeed, the quantity ⟨ Y^{2}_{500} ⟩ is equal to , and ⟨ L^{2}_{500} ⟩ is equal to . The crosscorrelation power spectrum of tSZ and Xray can be affected by the same effect. However, for a crosscorrelation this scatter bias depends on the correlation between the scatters of the M_{500} − Y_{500} and the M_{500} − L_{500} scaling relations. This question is addressed in Sect. 3.2.
We also need to have an estimate of the cluster temperature, T_{500}. In this work, we used the scaling law from Planck Collaboration XI (2011): (20)where the coefficients T_{⋆}, α_{T} and β_{T} are given in Table 1. This relation also presents an intrinsic scatter σ_{log T} = 0.14 ± 0.02. We verified that this scatter has no significant impact on the tSZX cross powerspectrum amplitude with respect to the scatter from M_{500} − Y_{500} and M_{500} − L_{500} relations.
2.6. Pressure and density profiles
The tSZ effect is directly proportional to the pressure integrated across the line of sight. In this work, we model the galaxy cluster pressure profile by a generalized Navarro Frenk and White (GNFW, Navarro et al. 1997; Nagai et al. 2007) profile of the form (21)For the parameters c_{500}, α, β, and γ, we used the bestfitting values from Arnaud et al. (2010) presented in Table 1. The absolute normalization of the profile P_{0} is set assuming the scaling laws Y_{500} − M_{500} presented in Sect. 2.5.
To model the Xray emission, we need both the density, n_{e}(r), and the temperature, T_{e}(r), profiles. Thus, we assume a polytropic equation of state (see, e.g., Komatsu & Seljak 2001), P(r) = n_{e}(r)T_{e}(r), with n_{e}(r) ∝ T_{e}(r)^{δ} where δ is the polytropic index. Considering that the Xray emission varies with and , the Xray brightness profile is proportional to P(r)^{ϵp}, where ranges from 1.5 to 2.0 for 1.0 < δ < ∞.
For the Xray flux to countrate conversion factor, we only assume an averaged temperature T_{500}, and then the Xray flux is directly proportional to . The overall normalization of the Xray emission profile is deduced from the scaling law L_{500} − M_{500} presented in Sect. 2.5.
Fig. 1 tSZXray crosscorrelation for the 1halo term in blue and for the 2halo term in red, considering the ROSAT hard band from 0.5 to 2.0 keV. The dashed lines represent the 1σ level considering variations in scalingrelation power laws. This crosscorrelation spectrum has been predicted using our fiducial cosmological model. 

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3. Results
3.1. The tSZX power spectrum
In Fig. 1, we present the angular cross power spectrum between tSZ and Xrays (assuming a 0.5−2.0 keV energy band for the ROSAT experiment). The power spectrum is predicted for our fiducial cosmological model. We observe that the tSZX power spectrum is dominated by the onehalo term throughout the entire range of multipole, from ℓ = 0 to ℓ = 10 000. For low multipoles (ℓ < 1000), the tSZX power spectrum follows a power law .
The correlation factor between tSZ and Xray surveys, , as a function of ℓ presents a slow variation from 0.8 at low ℓ to 0.6 at high ℓ. The smaller correlation factor on small scales highlight the difference in slopes in tSZ and Xray profiles in the core of the cluster, as the tSZ profile decreases with T_{e}n_{e} and the Xray profile with .
3.2. The scalingrelation scatter bias
The amplitude of the onehalo term of tSZX power spectrum is sensitive to the scatter of scaling relations, which produce an excess of power. The twohalo term is not affected by the scatter. For the tSZ autocorrelation power spectrum, the scatter produces a negligible bias of 0.5%. For the Xray power spectrum, the effect reach 3.3%.
The bias on tSZX cannot be estimated as easily. Indeed the correlation between M_{500} − Y_{500} and M_{500} − L_{500} scatters has to be known. The quantity ⟨ Y_{500}L_{500} ⟩ is equal to (1 + ρ_{YL}σ_{log Y}σ_{log L})Y_{500}L_{500}, where ρ_{YL} is the correlation between the M_{500} − Y_{500} and M_{500} − L_{500} scatters. Consequently, the bias is zero for a 0 correlation and maximum for a full correlation.
Using the Y_{500} − L_{500} scatter, σ_{log YL} = 0.14 ± 0.02 (Planck Collaboration XI 2011), it is possible to estimate ρ_{YL} through (22)This finding is consistent with an almost full correlation between M_{500} − Y_{500} and M_{500} − L_{500} scatters. This value for ρ_{YL} leads to a bias of 1.4% for the amplitude of the tSZX power spectrum. Finally, we note that the M_{500} − Y_{500} and M_{500} − L_{500} scatters account for the profile scatter from cluster to cluster.
3.3. Redshift and mass distribution of the tSZXray cross correlation
In Figs. 2 and 3, we present the distribution of the tSZXray cross correlation power spectrum as a function of the redshift and the mass for various values of ℓ from 20 to 10 000 for our fiducial model. We observe that the power below ℓ = 100 is dominated by a local object at redshifts below 0.2. Whereas at high multipole values, ℓ = 10 000, we are sensitive to structures up to z = 1.5. We observe that the small and large angular scales of the power spectrum sample distinct populations in terms of redshift.
Unlike the redshift distribution, we observe that the mass dependency presents small variations for ℓ ranging from 20 to 2000. For these multipoles, the power is dominated by objects with M_{500} above 10^{14}M_{⊙}. Lower mass objects only make significant contributions to the total power for a very high multipole value ℓ ≃ 10 000.
Fig. 2 Redshift distribution of the contribution to the total tSZ/Xray crosscorrelation power for several values of ℓ. In black for ℓ = 20, in dark blue for ℓ = 100, in light blue for ℓ = 200, in green for ℓ = 1000, in orange for ℓ = 2000, and in red for ℓ = 10 000. 

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Fig. 3 Mass distribution of the contribution to the total tSZ/Xray crosscorrelation power for several values of ℓ. In black for ℓ = 20, in dark blue for ℓ = 100, in light blue for ℓ = 200, in green for ℓ = 1000, in orange for ℓ = 2000, and in red for ℓ = 10 000. 

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The mass function predicts a number of objects above a given mass M_{500} that drastically increases when M_{500} decreases. Similarly, the comoving volume increases with increasing redshift (below z ≃ 2). The twohalo term presented in Sect. 2.3 is proportional to the number of clusters, which contributes significantly to the total power, to the square, unlike the onehalo terms that are linearly related to this quantity.
This explains the relative small amplitude of the twohalo term with respect to the onehalo term for the tSZX power spectrum. Indeed, the lowredshift depth and the high mass sensitivity of the tSZX power spectrum imply that the total power is dominated by a small number of objects.
Consequently, low values of α_{x} or α_{sz} promote the twohalo term with respect to the onehalo term. The twohalo term becomes significant at lowℓ for α_{x} + α_{sz} < 3. However, such values are excluded by existing constraints on the M_{500} − Y_{500} and the M_{500} − L_{500} relations (see Table 1 for the allowed uncertainty range and Fig. 1 for the impact on tSZX cross power spectrum of these uncertainties).
3.4. tSZX crosscorrelation dependencies with modeling parameters
Our modeling is affected both by cosmological and scaling law parameters. First, we focus on cosmological parameters, with a particular attention to H_{0}, Ω_{m}, and σ_{8}. In Fig. 4, we present the variation in the tSZX cross spectrum as a function of H_{0} from 60 to 80 km s^{1} Mpc^{1} in steps of 1 km s^{1} Mpc^{1}, Ω_{m} from 0.2 to 0.4 in steps of 0.01, and σ_{8} from 0.7 to 0.9 in steps of 0.01. In the most general case, those variations depend on the multipole ℓ, as presented in Figs. 2 and 3. Each multipole is sensitive to different regions of the mass function and thus present different sensitivities to the cosmological parameters: (23)where A_{cl} is the amplitude of the tSZX power spectrum. However as shown in Fig. 4, we do not observe a significant distortion of the shape of the crosscorrelation with a variation in the cosmological parameters.
Similar expressions can be used for tSZ and Xray autocorrelation spectra. In Table 2 we present the values of α_{H}(ℓ), α_{Ω}(ℓ), and α_{σ}(ℓ) for each spectra, tSZX, tSZauto, Xauto.
Fig. 4 Theoretical tSZXray crosscorrelation power spectra, as a function of, from top to bottom, H_{0}, Ω_{m}, and σ_{8}. From blue to red for 21 values of H_{0}, Ω_{m}, and σ_{8}, starting at 60 km s^{1} Mpc^{1}, 0.2, and 0.7 in steps of 1 km s^{1} Mpc^{1}, 0.01, and 0.01, respectively. 

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In Fig. 5, we present the power law index variations with respect to ℓ. For the tSZ autocorrelation we observe α_{H} ≃ 0; however, both the Xray autospectrum and the XtSZ cross spectrum present significant variations with H_{0}, with and . These dependency variation are produced by the L_{500} − M_{500} scaling relation. We also observe that α_{Ω} ranges from 2.8 to 3.8 and α_{σ} from 7 to 9. These variations in spectral indices with respect to ℓ are small.
In Table 2, we provide fitting formula for each power law index with respect to ℓ using the following parametric formula: (24)Values for p_{1}, p_{2}, p_{3}, p_{4} are provided in Table 2. We note that the power law indices are also functions of the cosmological parameters. Consequently, we stress that the formula given above has been estimated for cosmological parameters (H_{0}, Ω_{m}, σ_{8}) = (67,0.32,0.83), and thus can only be considered accurate for 10% variations around these parameters.
In addition to the sensitivity to cosmological parameters, the tSZX correlation is highly sensitive to the scaling relations described in Sect. 2.5 and cluster profiles described in Sect. 2.6, both of which are related. Consequently for the present work, we consider the profile from Arnaud et al. (2010) that has been used to estimate the Y_{500} − M_{500} scaling relation. However, we verified that the tSZX cross spectrum varies by 10% at most for various pressure profiles, as Arnaud et al. (2010), Planck Collaboration Int. V (2013), and Sayers et al. (2013) and for various polytropic indices from 1 to ∞ (isothermal case).
A variation in the scalingrelation normalization translates into a variation in the amplitude of the cross spectrum. However, the scaling law power law indices will produce a modification of the shape of the tSZX correlation.
In the following, we model the deviation from our reference scaling laws presented in Sect. 2.5 as where δα_{sz} and δα_{x} represent the deviations from the reference scaling law indices, α_{sz} and α_{x}, for M_{500} − Y_{500} and M_{500} − L_{500}, respectively.
In Fig. 6, we present the variation in the tSZX, Xray, and tSZ power spectra with the scaling law indices, α_{sz} and α_{x}, insteps of 0.025 for each index and 0.05 for their sum. We note that the tSZX power spectrum is only sensitive to the sum of the indices, α_{sz} + α_{x}. In terms of amplitude, the tSZX power spectrum follows (27)
Fig. 5 Power law index variation as a function of ℓ for in black the XtSZ crosscorrelation, in red the tSZ autocorrelation, and in blue the Xray autocorrelation. From top to bottom for α_{H}(ℓ), α_{Ω}(ℓ), and α_{σ}(ℓ). 

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Fig. 6 From top to bottom: variation in the theoretical tSZX, tSZtSZ, and XX power spectra as a function of α_{sz} + α_{x}, α_{sz} and α_{x}, in steps of 0.05, 0.025, and 0.025, respectively. 

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We observe in Fig. 6 that increasing the value of α_{sz} and α_{x} increases the power at lowℓ where highmass objets dominate the signal and decreases the power at high ℓ, that is dominated by lowmass objects. However, the shape distortion of the power spectra occurs at highℓ. The impact of scaling law indices start to be significant at ℓ > 1000, ℓ > 800, and ℓ > 2000 for tSZX, tSZ, and X power spectra, respectively. We observe that the Xray power spectrum presents the lowest sensitivity in terms of shape with respect to δα_{x}.
We can infer a global dependence of the tSZX crosscorrelation amplitude for fixed scalinglaw indices and fixed profiles, (28)where N is a normalization parameter for the mass function.
Modeling error budget for the tSZX cross spectrum and the X and SZ auto spectra.
3.5. Modeling uncertainties
Uncertainties on the predicted spectrum are produced by uncertainties on galaxies cluster properties and uncertainties on the cosmology. We used the uncertainties on cluster properties listed in Table 1 and propagated them to predict the power spectra. For the mass function, we assumed an overall uncertainty of 10% (Evrard et al. 2002) and for the bias, b, we assumed an uncertainty of 10% (e.g, Piffaretti & Valdarnini 2008). For the uncertainties on cosmological parameters, we consider two different sets of cosmological parameters, our fiducial model named cosmo 1, and a second model based on the best fit from Planck Collaboration XX (2014) named cosmo 2. For both sets of parameter, we propagated the uncertainties to the tSZX power spectra. We carefully account for correlation between uncertainties between parameters for each set.
We notice that the propagation of the uncertainty on α_{SZ,X} depends on the multipole, consequently we provide a range for the uncertainty for 0 < ℓ < 2000.
In Table 3, we present the modeling uncertainties on the tSZX cross correlation. We stress that these uncertainties translate into an overall normalization of the spectrum. We also provide the uncertainty levels for the tSZ and Xray auto correlation spectra. Assuming a fixed cosmology, we noted that the error budget is limited by our knowledge of b, leading to an uncertainty of about 35% on the amplitude of the tSX power spectrum. If we propagate the uncertainties on cosmological parameters to the tSZX power spectrum, we derived an uncertainty of 31% for the cosmo 1 and an uncertainty of 27% for cosmo 2. Cosmo 2 allows us to obtain slightly lower uncertainties for the tSZX prediction, because the degeneracy between cosmological parameters is similar for tSZ and tSZX spectra, unlike for cosmo 1 cosmology, which presents different degeneracies.
We finally note that for the tSZX cross power spectrum, the error budget from cluster properties and cosmological parameters have the same order of magnitude. The total uncertainty that includes both contributions reaches 48%.
4. Prediction of the tSZXray spectrum measurement
To estimate the tSZXray cross correlation, it is possible to use several approaches. One of them relies on the crosscorrelation of frequency maps of the microwave sky with an Xray map, see Sects. 4.3 and 4.4. Another one consists in using a recovered tSZ map (see, e.g., Hurier et al. 2013) and an Xray map, see Sect. 4.5. In the following, we discuss the advantages and drawbacks of each approach.
Measurement of the tSZX correlation is limited by both instrumental characteristics and contaminating astrophysical emissions. To estimate the constraints that can be reached on the tSZX power spectrum, we predicted the expected signaltonoise ratio assuming the Planck nominal mission characteristics (i.e., noise and beams Planck Collaboration I 2014) for the microwave observations and the RASS (Voges et al. 1999) for Xrays.
4.1. The microwave and Xray skies
Fig. 7 From left to right and top to bottom: predicted sensitivity to the tSZX cross correlation by computing the cross power spectra between the RASS hard band and the Planck channels from 70 to 545 GHz. The darkblue, blue, and lightblue shaded regions show the uncertainties levels at 1, 2, and 3 σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

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To simulate the sky emission at microwave frequencies with the appropriate level of noise, we used the Planck Sky Model (PSM, see Delabrouille et al. 2013, and references therein). At microwave frequencies, the main astrophysical emissions are the diffuse Galactic freefree, synchrotron, thermal dust emissions, the anomalous microwave emission, the emission from Galactic and extragalactic point sources, the CIB, the Zodiacal light emission, and the tSZ effect in clusters of galaxies.
To account for the signal from the Xray sky, we used the RASS data in the energy range [0.5, 2.0] keV, degraded at an angular resolution of FWHM = 2′, and projected again in the HEALpix pixelization (Górski et al. 2005) following a nearest neighbor interpolation. There are several types of astrophysical objects that emit in the Xrays: extragalactic ones such as galaxy clusters, black holes in AGN, the combination of unresolved Xray emitting objects producing the Xray background (Freyberg et al. 1992), but also galactic sources, mainly supernova remnants and stars.
In addition to the signal from galaxy clusters, there are other astrophysical emissions that are also correlated between the Xray and the microwave skies. This is the case of the radioloud AGNs and the CIB. Both AGN and CIB present a different frequency dependence than the tSZ effect and consequently can, in principle, be separated from the galaxy clusters’ contribution to the tSZX cross correlation. In addition, AGNs are pointlike at Planck and ROSAT angular resolutions, and thus can be separated from the clusters’ contribution by the shape of the power spectrum.
The emission from our galaxy is also present on sky at microwave frequencies (synchrotron, FreeFree, and thermal dust) and in ROSAT energy bands (n_{H} absorption and galactic Xray emissions). However, considering only the ROSAT hard band (0.5−2.0 keV) reduces the effect of the n_{H} absorption, and utilizing a galactic mask reduces the contamination by galactic foregrounds. As for the contamination by extragalactic point sources, such contamination will present a frequency dependence that differs from a tSZ spectrum, and thus can be distinguished using multifrequency analysis.
4.2. Statistical and systematic uncertainties
We can estimate the statistical uncertainties, on the tSZX correlation, using

our prediction for a tSZX crosscorrelation power spectrum from the galaxy cluster, ;

our prediction for tSZ and Xray autocorrelation power spectra from the galaxy cluster, and ;

the measured crosscorrelation, , between the microwave sky from the PSM, denoted T _{ ν } for frequency ν , and the Xray sky from RASS data, denoted R;

the measured auto correlations, and , of the microwave and Xray skies;

the measured cross correlation, , between the tSZ map constructed from the PSM, denoted y _{PSM} , and the Xray sky from RASS data, denoted R;

the measured auto correlation, of the tSZ map.
The expression uncertainties for each tSZX detection method is presented in Sects. 4.3, 4.4, and 4.5.
Our estimation of uncertainties through simulations of the microwave sky does not allows an estimate of the systematic uncertainties. To account for systematics uncertainties, we considered three cases for describing the measured tSZX cross correlation.

Case 1: considering only the contribution of galaxy clusters to thetSZX correlation.

Case 2: considering the contributions from galaxy clusters and AGNs to tSZX correlation.

Case 3: considering galaxy clusters, AGNs, and CIBX contributions.
The complete description of the cross power spectrum for Case 3 reads as (29)with F_{CIB} the CIB SED (Gispert et al. 2000), F_{AGN}(ν) a typical radioloud AGN SED assuming a spectral index of −0.7 in intensity units, A_{cl} (see Eq. (28)), A_{CIB}, and A_{AGN} are the parameters of the model and account for galaxy clusters, CIBX, and AGN contributions to the tSZX cross power spectrum, respectively. Case 1 assumes A_{CIB} = 0 and A_{AGN} = 0 and Case 2 assumes A_{CIB} = 0.
This modeling assumes that the CIBX correlation presents a similar shape, as a function of ℓ, to the clusters’ contribution to the tSZX cross spectrum and assumes that the AGN contribution is Poissonian, since the AGN clustering can be neglected.
In the following, we consider the multipole range 40 < ℓ < 2000 for our signaltonoise ratio prediction.
4.3. Cross correlation spectrum from frequency maps
The tSZX cross correlation can be directly estimated from the correlation between Xray maps and microwave fullsky observations at a given frequency, denoted . We estimate the expected level of uncertainties when correlating the RASS hardband and microwave maps at frequency ν. We mask 30% of the sky by applying a cut on the thermal dust emission intensity. Then, we estimate the uncertainties following (29)where is the uncertainties on the cross power spectrum between microwave and Xray skies, and f_{sky} is the sky fraction used for the analysis. The first term in Eq. (29) corresponds to the cosmic variance of the tSZX cross correlation, and the other terms account for the uncertainties produced by foreground emissions.
In Fig. 7, we present the resulting uncertainty level at 1, 2, and 3σ as a function of the frequency. We also present the expected absolute value of the tSZX cross correlation for our fiducial model. All spectra are displayed in units of K_{CMB} cts s^{1} arcmin^{2}sr. Each spectrum has been corrected for the mask effect (see Tristram et al. 2005) and the beam effects. We chose to present the uncertainty for multipole bins with Δℓ = 40.
Detection signaltonoise of the tSZX cross power spectrum frequency per frequency from 30 to 857 GHz considering values of ℓ from 40 to 2000.
For the lowest frequencies, below 70 GHz, the signal is completely dominated by the instrumental noise contribution. For intermediate frequencies from 70 to 217 GHz, the main uncertainty is the CMB contamination, and the uncertainty level clearly shows the CMB features, mainly the first three acoustic pics. Above 353 GHz, the uncertainties are dominated by the thermal dust contamination.
We note that at 217 GHz the tSZ emission is not rigorously zero; however, the tSZ transmission in this channel is faint (Hurier et al. 2014). Consequently, this channel can be used to check systematic effects.
In Table 4, we present the expected signaltonoise for tSZX correlation as a function of frequency. The signaltonoise is provided assuming Cases 1 and 2. We observe that in Case 3, we reach a signaltonoise above six for only two channels, 100 and 143 GHz. However, the main limitation in that case is the astrophysical emission from the microwave sky, such as CMB and thermal dust emission. These emissions are correlated from one frequency to another.
4.4. Cross power spectrum from cleaned frequency maps
Fig. 8 From left to right and top to bottom: predicted sensitivity to the tSZX cross correlation for the cleaned cross power spectra between the RASS hard band and the Planck channels at 70, 100, 143, 353, and 545 GHz, respectively. The darkblue, blue, and lightblue shaded regions show the uncertainty levels at 1, 2, and 3σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

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Detection signaltonoise of the tSZX cross power spectrum frequency per frequency after cleaning at 70, 100, 143, 353, and 545 GHz considering values of ℓ from 40 to 2000.
To increase the signal to noise of the tSZX detection, we combine the different frequencies to remove the contribution from CMB and thermal dust. This cleaning is performed by subtracting the 217 GHz spectrum to other spectra to remove CMB contamination and decorrelating each channel from the 857 GHz map to reduce thermal dust contamination. (31)where ρ_{ν} is the correlation factor between the map at the frequency ν and the map at 857 GHz. This factor is computed on the area of the sky that is not masked. We computed these cleaned angular power spectra at 70, 100, 143, 353, and 545 GHz.
We propagate the uncertainties considering the correlation between cross spectra, (32)
In Fig. 8, we present the obtained power spectrum. We observe that the uncertainties at lowℓ are dominated by foreground residuals. Indeed, the decorrelation from the 857 GHz assuming a single scaling coefficient ρ_{ν} does not account for thermal dust SED variation across the sky and thus leads to residual emission that dominates the uncertainties on those scales. At highℓ the uncertainties are dominated by the instrumental noise. In Fig. 8, we also observe that our cleaning is particularly efficient on intermediate scales from ℓ = 100 to ℓ = 1000, owing to CMB contamination removal.
In Table 5 we present the expected signaltonoise of the tSZX cross power spectrum signal in the multipole range 40 < ℓ < 2000. We provide the results in the three cases described in Sect. 4.2.
For Case 2, we performed the adjustment of A_{AGN} individually per frequency. For Case 3, we performed the estimation of the expected signaltonoise ratio per frequency and considered all frequencies. We note that for both adjustments, the parameters A_{CIB} and A_{AGN} are fit considering all frequencies. This adjustment is performed considering the global covariance matrix of all spectra.
This procedure explains the increase in signaltonoise between Cases 2 and 3. As for Case 3, we consider the multifrequency information of the tSZX cross correlation for the fit of our foreground model.
In a realistic case, Case 3, we reach a signaltonoise of 13.4 for the tSZX cross correlation at 143 GHz. Considering all five frequencies, we obtain a global signaltonoise of 17.5.
4.5. Crosscorrelation from Compton parameter maps
We construct a tSZ map from the PSM simulations of microwave sky observations from 100 to 857 GHz using the MILCA method (Hurier et al. 2013). Then, we estimate the uncertainties on the cross power spectrum, , between a tSZ map and an Xray map as (33)where y_{PSM} corresponds to the tSZ map constructed from the PSM simulations.
In Fig. 9, we present the obtained levels of uncertainties. The tSZX power spectrum for our fiducial model is above 2σ for each bin of Δℓ = 40 from ℓ = 40 to 2000. The main limitations are the instrumental noise and CIB residuals that cannot be removed by a linear combination. If we do not consider contamination by correlated astrophysical emissions such as radioloud AGNs and CIB, we obtain an overall signaltonoise of 62.3 from ℓ = 40 to ℓ = 2000. If we consider contamination by AGNs and CIB we obtain a signaltonoise of 31.5.
We note in Sect. 3.5 that the modeling derived from present constraints leads to about a 48% uncertainty on the amplitude of the tSZX cross correlation. Considering Case 3 and a signaltonoise of 31.5, the amplitude of the tSZX crosscorrelation can be obtained at 3.2% precision. As a consequence, the utilization of a tSZ map allows to set the tighter constraints on the tSZX crosscorrelation and to increase our knowledge of cosmological and astrophysical parameters in the related degeneracy space by a factor of 15.
4.6. Constraints on astrophysical and cosmological parameters
The amplitude of the tSZX cross correlation can be constraints at a precision of 3.2%. However, this amplitude is sensitive to several parameters from both cosmology and scaling laws, see Eq. (28). The expected constraints on the tSZX cross correlation normalization reads (34)in the case of the correlation between a tSZ map and an Xray map considering the contamination by both AGN and CIB emissions.
To obtain the uncertainty over a single parameter, it is necessary for it to carefully propagate the uncertainties from other parameters considering the global covariance matrix of all parameters. Beyond the amplitude of the tSZX crosscorrelation, if we fix the profiles of galaxy clusters, the expected high level of significance of the tSZX cross correlation should allow us to constrain the scalinglaw spectral indices.
Fig. 9 Predicted sensitivity to the tSZX crosscorrelation for the cross power spectra between the RASS hard band and a MILCA tSZmap. The darkblue, blue, and lightblue shaded regions show the uncertainties levels at 1, 2, and 3σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

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Fig. 10 Likelihood function as a function of the deviation from scaling law indices δα_{sz} and δα_{x} derived from tSZ autocorrelation power spectrum and tSZX cross power spectrum. 

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In Fig. 10, we present the constraints from our simulated data. The tSZX cross correlation is only sensitive to the sum δα_{x} + δα_{sz} for fixed density and temperature profiles. To distinguish the two indices, we used the tSZ autocorrelation power spectrum assuming uncertainty levels from Planck Collaboration VIII (2011).
This measurement is limited by the multipole range accessible, for example, by Planck and ROSAT, ℓ ≤ 2000. In that case, we obtained Δ(δα_{sz}) = 0.10 and Δ(δα_{sz}) = 0.16 for the scaling law index deviation parameters. If we add a measurement of tSZ spectrum at ℓ = 3000, considering tSZ data at higher resolution from SPTlike (Shirokoff et al. 2011) or ACTlike (Sievers et al. 2013) experiments, we can reach Δ(δα_{sz}) = 0.08 and Δ(δα_{x}) = 0.14. These constraints on the scalinglaw slopes with respect to the cluster mass are competitive with constraints from the scalinglawdedicated analyses presented in Table 1.
Finally, we estimated the degeneracy between profile and scalingrelation parameters considering the tSZX cross power spectrum. For small variations (up to 10%) around the Arnaud et al. (2010) profile, we can express the degeneracy relation as (35)From this result, we observed that the shape of the tSZX cross spectrum is highly sensitive to both scalingrelation and profile parameters. Consequently, the tSZX spectrum is a powerful probe to simultaneously test the combination of scaling laws and profiles.
5. Conclusion
We have presented a complete uptodate modeling of the tSZ, Xray, and tSZX power spectra. We have carefully studied the sensitivity to cosmological and astrophysical parameters, completing a previous analysis on the topic (Diego et al. 2003). For the variations in the tSZX crosscorrelation, tSZ, and Xray autocorrelation with H_{0}, Ω_{m}, and σ_{8}, we provided accurate analytical fitting formulae in the range 0 < ℓ < 10 000.
Then, we carefully propagated the uncertainties on the cosmological and scaling law parameters of our modeling to the predicted power spectra, leading to an overall uncertainty on the normalization of the tSZX power spectrum by about 48%. This result highlights our limited knowledge of this cosmological probes.
We note that the contributions to the total uncertainty from clusters’ scaling relations and cosmological parameters are at same order of magnitude. The main contributions to the total uncertainty are produced by the bias b, 35%, and cosmological parameters, 27%. These large uncertainties illustrate the importance of an accurate measurement of the tSZX cross angular spectrum to set constraints on both cluster properties and cosmological parameters.
We stress that most of our modeling parameters act on the tSZX power spectrum as an overall amplitude factor. This leads to high degeneracy between those parameters, and thus the tSZX cross correlation needs to be used in addition of other probes to break the degeneracies. We note that our prediction cannot be directly compared to the measurement recently performed by Hajian et al. (2013), since their measurement concerned the crosscorrelation between tSZ maps and Xray catalogs, and consequently, it presents difference dependencies with cosmological parameters.
We predicted the expected signaltonoise that can be reached using a simulated microwave sky and ROSAT data for the Xray sky. We considered three approaches to extracting the tSZX cross power spectrum. We demonstrate that in the case of the crosscorrelation between a tSZ map and an Xray map, we reach a signaltonoise of 31.5. In this case, we can reach a measurement of the tSZX cross correlation amplitude at about 3%, improving the actual constraints (from our knowledge of cosmological parameters and scaling laws) on the predicted spectrum by about a factor of 10.
We also studied the possibility of constraining the slope of the tSZ and Xray scaling laws and the parameters of a GNFW profile using the shape of the tSZX cross spectrum. We conclude that scalinglaw and profile parameters are highly degenerated. Thus, constraints from tSZX spectrum allows the global consistency of scaling laws and profiles to be checked.
Future experiment at high resolution and high sensitivity for tSZ surveys and Xray measurements will, in the near future, allow the increase of the expected constraints, especially by providing a wider range of multipoles, higher sensitivity, and more frequencies.
Acknowledgments
We acknowledge the support of the French Agence Nationale de la Recherche under grant ANR11BD56015. We are grateful to an anonymous referee for his/her comments and suggestions. We also thank M. Arnaud, J. B. Melin, E. Pointecouteau, and G. Pratt for useful discussions related to this work. This research has made use of the ROSAT allsky survey data which have been processed at MPE. We acknowledge the use of HEALPix package (Górski et al. 2005).
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All Tables
Scalinglaw parameters and error budget for both Y_{500} − M_{500} (Planck Collaboration XX 2014), L_{500} − M_{500} (Arnaud et al. 2010), and Y_{500} − T_{500} (Planck Collaboration XX 2014) relations.
Modeling error budget for the tSZX cross spectrum and the X and SZ auto spectra.
Detection signaltonoise of the tSZX cross power spectrum frequency per frequency from 30 to 857 GHz considering values of ℓ from 40 to 2000.
Detection signaltonoise of the tSZX cross power spectrum frequency per frequency after cleaning at 70, 100, 143, 353, and 545 GHz considering values of ℓ from 40 to 2000.
All Figures
Fig. 1 tSZXray crosscorrelation for the 1halo term in blue and for the 2halo term in red, considering the ROSAT hard band from 0.5 to 2.0 keV. The dashed lines represent the 1σ level considering variations in scalingrelation power laws. This crosscorrelation spectrum has been predicted using our fiducial cosmological model. 

Open with DEXTER  
In the text 
Fig. 2 Redshift distribution of the contribution to the total tSZ/Xray crosscorrelation power for several values of ℓ. In black for ℓ = 20, in dark blue for ℓ = 100, in light blue for ℓ = 200, in green for ℓ = 1000, in orange for ℓ = 2000, and in red for ℓ = 10 000. 

Open with DEXTER  
In the text 
Fig. 3 Mass distribution of the contribution to the total tSZ/Xray crosscorrelation power for several values of ℓ. In black for ℓ = 20, in dark blue for ℓ = 100, in light blue for ℓ = 200, in green for ℓ = 1000, in orange for ℓ = 2000, and in red for ℓ = 10 000. 

Open with DEXTER  
In the text 
Fig. 4 Theoretical tSZXray crosscorrelation power spectra, as a function of, from top to bottom, H_{0}, Ω_{m}, and σ_{8}. From blue to red for 21 values of H_{0}, Ω_{m}, and σ_{8}, starting at 60 km s^{1} Mpc^{1}, 0.2, and 0.7 in steps of 1 km s^{1} Mpc^{1}, 0.01, and 0.01, respectively. 

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In the text 
Fig. 5 Power law index variation as a function of ℓ for in black the XtSZ crosscorrelation, in red the tSZ autocorrelation, and in blue the Xray autocorrelation. From top to bottom for α_{H}(ℓ), α_{Ω}(ℓ), and α_{σ}(ℓ). 

Open with DEXTER  
In the text 
Fig. 6 From top to bottom: variation in the theoretical tSZX, tSZtSZ, and XX power spectra as a function of α_{sz} + α_{x}, α_{sz} and α_{x}, in steps of 0.05, 0.025, and 0.025, respectively. 

Open with DEXTER  
In the text 
Fig. 7 From left to right and top to bottom: predicted sensitivity to the tSZX cross correlation by computing the cross power spectra between the RASS hard band and the Planck channels from 70 to 545 GHz. The darkblue, blue, and lightblue shaded regions show the uncertainties levels at 1, 2, and 3 σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

Open with DEXTER  
In the text 
Fig. 8 From left to right and top to bottom: predicted sensitivity to the tSZX cross correlation for the cleaned cross power spectra between the RASS hard band and the Planck channels at 70, 100, 143, 353, and 545 GHz, respectively. The darkblue, blue, and lightblue shaded regions show the uncertainty levels at 1, 2, and 3σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

Open with DEXTER  
In the text 
Fig. 9 Predicted sensitivity to the tSZX crosscorrelation for the cross power spectra between the RASS hard band and a MILCA tSZmap. The darkblue, blue, and lightblue shaded regions show the uncertainties levels at 1, 2, and 3σ for multipole bins Δℓ = 40. The absolute value of the theoretical tSZX cross angular power spectrum is displayed as a red line. 

Open with DEXTER  
In the text 
Fig. 10 Likelihood function as a function of the deviation from scaling law indices δα_{sz} and δα_{x} derived from tSZ autocorrelation power spectrum and tSZX cross power spectrum. 

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In the text 
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