Predicting the CIBφ contamination in the crosscorrelation of the tSZ effect and φ
Institut d’Astrophysique Spatiale, CNRS (UMR 8617) and Université ParisSud
11,
Bâtiment 121,
91405
Orsay,
France
email:
ghurier@ias.upsud.fr
Received: 21 January 2015
Accepted: 6 February 2015
The recent release of Planck data gives access to a full sky coverage of the thermal SunyaevZel’dovich (tSZ) effect and of the cosmic microwave background (CMB) lensing potential (φ). The crosscorrelation of these two probes of the largescale structures in the Universe is a powerful tool for testing cosmological models, especially in the context of the difference between galaxy clusters and CMB for the bestfitting cosmological parameters. However, the tSZ effect maps are highly contaminated by cosmic infrared background (CIB) fluctuations. Unlike other astrophysical components, the spatial distribution of CIB varies with frequency. Thus it cannot be completely removed from a tSZ Compton parameter map, which is constructed from a linear combination of multiple frequency maps. We have estimated the contamination of the CIBφ correlation in the tSZφ powerspectrum. We considered linear combinations that reconstruct the tSZ Compton parameter from Planck frequency maps. We conclude that even in an optimistic case, the CIBφ contamination is significant with respect to the tSZφ signal itself. Consequently, we stress that tSZφ analyses that are based on Compton parameter maps are highly limited by the bias produced by CIBφ contamination.
Key words: galaxies: clusters: general / cosmological parameters / largescale structure of Universe / cosmic background radiation / galaxies: clusters: intracluster medium / infrared: diffuse background
© ESO, 2015
1. Introduction
Modern cosmology extensively used cosmic microwave background (CMB) data. During their propagation along the line of sight, photons are affected by several physical processes such as the thermal SunyaevZel’dovich effect (tSZ, Sunyaev & Zeldovich, 1969, 1972) and CMB gravitational lensing (Blanchard & Schneider, 1987), which trace the gravitational potential integrated along the line of sight, φ.
At microwave frequencies, foreground emissions contribute to the total signal of the sky, for example the cosmic infrared background (CIB; Puget et al., 1996; Fixsen et al., 1998). The tSZ effect, gravitational lensing, and the CIB are tracers of the largescale structures in the matter distribution of the Universe. They have been powerful sources of cosmological and astrophysical constraints (see e.g., Planck Collaboration XX, 2014; Planck Collaboration XXX, 2014; Planck Collaboration XVII, 2014).
All these probes present correlations through their relation to the largescale distribution of matter in the Universe. The CIBφ correlation has already been detected with high significance (Planck Collaboration XVIII, 2014). More recently, the tSZφ correlation estimated from a Compton parameter map (Hill & Spergel, 2014) has been used to constrain cosmological parameters.
We discuss the contamination from the CIBφ correlation into the tSZφ correlation estimated from a Compton parameter map. In Sect. 2, we present a coherent modelling of the tSZ, φ, and the CIB spectra and crosscorrelation. Then in Sect. 3, we estimate the CIBφ contamination in tSZφ crosscorrelation. In Sect. 4, we discuss the cleaning process proposed by Hill & Spergel (2014) and estimate the residual contamination level. Finally in Sect. 5, we conclude.
Throughout the paper, we consider H_{0} = 67.1, σ_{8} = 0.80 and Ω_{m} = 0.32 as our fiducial cosmological model, unless otherwise specified.
2. Modelling the tSZ, φ, and CIB crosscorrelations
The total powerspectrum between the A and B quantities (tSZ, φ or CIB) can be expressed as (1)with the Poissonian contribution and the contribution from large angular scale correlations.
2.1. Poissonian term
The Poissonian or onehalo term of the powerspectrum can be written using a halo model as the sum of the contributions from each halo. The number of halos per unit of mass and redshift, , is given by the mass function (see e.g., Tinker et al., 2008).
Using the limber approximation, we can write the onehalo term as (2)The tSZ contribution can be written as (3)with Y_{500} the tSZ flux of the clusters, related to the mass, M_{500} via the scaling law presented in Eq. (4), and y_{ℓ} the Fourier transform on the sphere of the cluster pressure profile from Planck Collaboration XX (2014) per unit of tSZ flux.
We used the M_{500} − Y_{500} scaling law presented in Planck Collaboration XX (2014), (4)with E(z) = Ω_{m}(1 + z)^{3} + Ω_{Λ}. The coefficients Y_{⋆}, α_{sz} and β_{sz}, taken from Planck Collaboration XX (2014), are given in Table 1. The mean bias, (1 − b), between Xray mass and the true mass was estimated from numerical simulations, b ≃ 0.2 (see Planck Collaboration XX, 2014, for a detailed discussion of this bias).
Cosmological and scalinglaw parameters for our fiducial model for both Y_{500} − M_{500} (Planck Collaboration XX, 2014) and the L_{500} − M_{500} relation fitted on CIB spectra.
The lensing contribution can be written as (5)with χ the comoving distance, χ′ the comoving distance of the CMB, and ψ_{ℓ} the 3D lensing potential Fourier transform on the sky.
We can express the potential ψ as a function of the density contrast, (6)with a the universe scale factor and δ the density contrast. Then, the lensing contribution reads (7)where δ_{ℓ} is the Fourier transform of the density contrast profile.
The CIB contribution can be written as (8)with , the infrared flux at frequency ν of the host halo and c_{ℓ} the Fourier transform of the infrared profile.
To model the L_{500} − M_{500} relation we used a parametric relation derived from the relation proposed in Shang et al. (2012). To compute the tSZCIB correlation, we need to relate tSZ halos with CIB emissions. Consequently, we cannot use a parametrization that relates galaxy halos with the CIB flux. We need a relation that relates the galaxy cluster halo to the associated total CIB flux.
We chose to parametrize the CIB flux, L_{500}, at galaxy cluster scale with a power law of the mass^{1}, M_{500}, (9)where L_{0} is a normalization parameter, T_{d}(z) = T_{0}(1 + z)^{αcib} is the thermal dust temperature, and Θ[ν,T_{d}] is the typical spectral energy distribution (SED) of a galaxy that contributes to the total CIB emission,
with ν_{0} the solution of . The coefficients T_{0}, α_{cib}, β_{cib}, γ_{cib}, δ_{cib}, and ϵ_{cib} are given in Table 1.
2.2. Largescale correlation terms
We now focuss on the contribution from large angular scale correlations in the matter distribution to the tSZφCIB crosscorrelation powerspectra matrix. We expres this contribution as (10)with P_{k}, the matter powerspectrum.
In the following subsections we present the window functions, W, related to each contribution.
For the tSZ effect, we considered the twohalo largescale correlation. We neglected the contribution produced by diffuse tSZ emission from the warm hot interstellar medium. The twohalo contribution reads (11)where b_{lin} is the bias relating the halo distribution to the overdensity distribution. We considered the bias from Mo & White (1996) and Komatsu & Kitayama (1999), which is realistic at galaxy cluster scale.
For the lensing potential, we accounted for the largescale correlation by considering structures in the linear growth regime as(12)For the CIB at frequency ν, we considered the twohalo largescale correlation as (13)It is worth noting that the CIBφ crosscorrelation is dominated by the twohalo term^{2}, in contrast to the tSZCIB and tSZφ correlations which present a significant contribution from both one and twohalo terms.
Fig. 1 Distributions of the tSZ, φ, and CIB power as a function of the redshift at ℓ = 1000. The black solid line depicts the lensing potential, φ, the grey dashed line the tSZ effect, the dark blue, light blue, green, yellow, orange, and red dashed lines plot the CIB at 100, 143, 217, 353, 545, and 857 GHz, respectively. 

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2.3. Redshift distribution
In Fig. 1, we present the redshift distribution of power for the tSZ, CIB, and lensing potential autocorrelation spectra. The tSZ effect is produced by structures at low redshift (z< 2). The CIB and φ spectra are dominated by objects at higher redshift.
The mean redshift of structures that dominate in the CIB is a function of the considered frequency. These redshift windows highlight the high degree of correlation between the CIB and φ and the low degree of correlation between tSZ and the other two probes of largescale structures.
For example, the high degree of correlation between the CIB and φ have recently been used to detect the lensing B modes of polarization by the SPT collaboration (Hanson et al., 2013).
Consequently, small CIB residuals in a tSZ Compton parameter map can lead to a significant bias in the tSZφ correlation analysis.
3. CIBφ contamination in tSZφ crosscorrelation
3.1. tSZ Compton parameter map
The recently released Planck data allow constructing fullsky tSZ Compton parameter map (Planck Collaboration XXI, 2014) using internal linear combination (ILC) component separation methods (see e.g., Remazeilles et al., 2011; Hurier et al., 2013). These methods reconstruct the tSZ signals using linear combinations (14)where T_{ν} is the intensity at the frequency ν and w_{ν} are the weights of the linear combination. For ILCbased methods, these weights are computed by minimizing the variance of the reconstructed ymap under constraints.
The level of CIBφ contamination in a tSZφ analysis based on a tSZ Compton parameter map depends on the weights, w_{ν}, used for the linear combination.
We here considered the weights provided in Table 2 of Hill & Spergel (2014), which have been used to constrain the cosmology from the tSZφ crosscorrelation powerspectrum.
Fig. 2 Residual contamination before cleaning in light blue and after cleaning in dark blue. The red solid line presents the tSZφ correlation for our fiducial model. The solid blue lines are the contamination considering the set of weights from Hill & Spergel (2014) for f_{sky} = 0.3. The dashed blue lines are the contamination considering the set of weights from Hill & Spergel (2014) for f_{sky} = 0.2. 

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3.2. Propagation of CIBφ spectra
We stress that the CIB is composed of different populations of objects at different frequencies. The associated powerspectra present variations of shape with respect to the frequency. The CIBφ spectra present the same variations of shape with frequency. The CIBφ contamination in the tSZφ crosscorrelation reads (15)In Fig. 2, we present our estimation of the CIBφ contamination considering the set of weights from Hill & Spergel (2014) and compare it with the expected tSZφ crosscorrelation powerspectrum for our fiducial model.
The contamination is high, with a similar amplitude as the tSZφ correlation. Furthermore, the contamination level is highly dependent on the set of weights. For different sets of weights, the contamination varies an the same order of magnitude as the tSZφ signal.
This variability highlights the difficulty of controlling the CIBφ contamination that can lead to a high bias in tSZφ analysis.
However, the contamination level is highly dependent on the exact CIB frequency correlation matrix. Consequently, the derived value has to be considered as an orderofmagnitude estimate of the effective contamination.
4. Cleaning the CIBφ contamination
In their analysis, Hill & Spergel (2014) proposed a method of cleaning for CIBφ contamination in tSZφ spectrum. Their method consists of a correction of the form (16)where is the measured correlation between the tSZ and potential maps. The factor α_{corr} is computed from an adjustment of on the crosscorrelation of the tSZ Compton parameter and the 857 GHz maps.
We propose to test the accuracy of this procedure using our modelling of the tSZ, φ, and CIB crosscorrelations. In their analysis Hill & Spergel (2014) neglected the tSZCIB correlation contribution for computing α_{corr}. However, at the concerned frequencies, the tSZCIB correlation factor can reach about 20% and therefore cannot be neglected. Neglecting tSZCIB correlation for the cleaning process leads to an underestimation of the CIBφ contamination. In an optimistic case^{3}, α_{corr} reads (17)We computed this value for each set of weights at ℓ = 600. The value of α_{corr} varies by 3% depending on the value of ℓ used for the computation (300 <ℓ< 1500), with a higher level of contamination using higher ℓ values. Hill & Spergel (2014) found for f_{sky} = 0.3. Considering the associated set of weight we predict for 300 <ℓ< 1500. With this, we estimate the associated residual contamination in the tSZφ spectrum as (18)In Fig. 2, we present the obtained residuals for the set of weights from Hill & Spergel (2014). The CIBφ residual contributes with (20 ± 10)% of the tSZφ signal for ℓ ranging from 500 to 2000.
This level of contamination is dependent on the CIB luminosity function (see Sect. 2.2). The uncertainty level was estimated by propagating the CIB luminosity function uncertainties (see Planck Collaboration XXX, 2014, for details on estimating the CIB luminosity function parameters). This uncertainty level is dominated by δ_{cib} and ϵ_{cib}, which set the amount of correlation between the CIB maps observed at different frequencies. Other parameters, such as b_{lin} or CIB SED parameters, have a weaker effect on the CIBφ residual estimation.
5. Conclusion and discussion
We have presented a modelling of the tSZφCIB crosscorrelations. Based on this modelling, we predicted the expected CIBφ contamination in the tSZφ powerspectra deduced from a Compton parameter map built by linearly combining Planck channels from 100 to 857 GHz.
We demonstrated that the expected level of contamination from the CIBφ is at the same level as the tSZφ signal itself.
We also demonstrated that the CIBφ contamination level is highly dependent on the set of weights used for constructing of the Compton parameter map. To do this, we used realistic values deduced from the sky, for the weights.
We tested a cleaning method for this bias and showed that the level of residual bias can reach about 20% of the tSZφ spectrum
for the used set of weights. Consequently, we stress that an tSZφ analysis that is based on a multifrequency Compton parameter map may present a high level of bias. A careful CIBφ analysis has to be performed simultaneously to a tSZφ analysis to avoid high bias in the final results.
Hill & Spergel (2014) constrained σ_{8}(Ω_{m}/ 0.282)^{0.26} = 0.824 ± 0.029 and found that the tSZφ powerspectrum scales as . Thus, a bias of (20 ± 10)% on the the tSZφ amplitude produces a bias of (3.3 ± 1.7)% on the σ_{8} amplitude. As a consequence, the CIBφ induced residual bias in this measurement contributes about 1σ of the uncertainty. Such a bias may explain the difference between the tSZφ crosscorrelation and cosmological parameter estimation made with a tSZ spectrum. The latter favours lower values for σ_{8}, with σ_{8}(Ω_{m}/ 0.28)^{0.395} = 0.784 ± 0.016.
Acknowledgments
We are grateful to N. Aghanim and M. Douspis for useful discussions. We acknowledge the support of the French Agence Nationale de la Recherche under grant ANR11BD56015.
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All Tables
Cosmological and scalinglaw parameters for our fiducial model for both Y_{500} − M_{500} (Planck Collaboration XX, 2014) and the L_{500} − M_{500} relation fitted on CIB spectra.
All Figures
Fig. 1 Distributions of the tSZ, φ, and CIB power as a function of the redshift at ℓ = 1000. The black solid line depicts the lensing potential, φ, the grey dashed line the tSZ effect, the dark blue, light blue, green, yellow, orange, and red dashed lines plot the CIB at 100, 143, 217, 353, 545, and 857 GHz, respectively. 

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In the text 
Fig. 2 Residual contamination before cleaning in light blue and after cleaning in dark blue. The red solid line presents the tSZφ correlation for our fiducial model. The solid blue lines are the contamination considering the set of weights from Hill & Spergel (2014) for f_{sky} = 0.3. The dashed blue lines are the contamination considering the set of weights from Hill & Spergel (2014) for f_{sky} = 0.2. 

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In the text 