Constraining galaxy cluster velocity field with the thermal SunyaevZel’dovich and kinematic SunyaevZel’dovich crossbispectrum
^{1} Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza de San Juan, 1, planta 2, 44001, Teruel Spain
email: ghurier@cefca.es
^{2} Institut d’Astrophysique Spatiale, CNRS (UMR 8617) Université ParisSud 11, Bâtiment 121, 91405 Orsay, France
Received: 11 September 2016
Accepted: 11 March 2017
The SunyaevZel’dovich (SZ) effects are produced by the interaction of cosmic microwave background (CMB) photons with the ionized and diffuse gas of electrons inside galaxy clusters integrated along the line of sight. The two main effects are the thermal SZ (tSZ) produced by thermal pressure inside galaxy clusters and the kinematic SZ (kSZ) produced by peculiar motion of galaxy clusters compared to CMB restframe. The kSZ effect is particularly challenging to measure as it follows the same spectral behavior as the CMB, and consequently cannot be separated from the CMB using spectral considerations. In this paper, we explore the feasibility of detecting the kSZ through the computation of the tSZCMBCMB crosscorrelation bispectrum for current and future CMB experiments. We conclude that the next generation of CMB experiments will offer the possibility to detect the tSZkSZkSZ bispectrum at high signaltonoise ration (S/N). This measurement will constraints the intracluster dynamics and the velocity field of galaxy cluster that is extremely sensitive to the growth rate of structures and thus to dark energy properties. Additionally, we also demonstrate that the tSZkSZkSZ bispectrum can be used to break the degeneracies between the massobservable relation and the cosmological parameters to set tight constraints, up to 4%, on the Y − M relation calibration.
Key words: largescale structure of Universe / cosmic background radiation / cosmological parameters / galaxies: clusters: intracluster medium / galaxies: clusters: general
© ESO, 2017
1. Introduction
Present cosmic microwave background (CMB) experiments such as ACTT (Sievers et al. 2013), SPT (Bleem et al. 2015), and Planck (Planck Collaboration I 2016) have mapped the CMB primary temperature anisotropies with unprecedented precision, and constrained cosmological parameters from the CMB angular powerspectrum analyses (see e.g., Sievers et al. 2013; Planck Collaboration XIII 2016; de Haan et al. 2016). However, the measurement of secondary anisotropies, such as the SunyaevZel’dovich (SZ) effects (Sunyaev & Zeldovich 1972), is still limited by the noise level and angular resolution in current experiments.
Thanks to it’s spectral behavior, the tSZ effect can be isolated from the CMB and foreground emissions (Remazeilles et al. 2011; Hurier et al. 2013). The thermal SZ (tSZ) effect, produced by thermal electron in intracluster medium, has been shown as a powerful probe to detect galaxy clusters (Hasselfield et al. 2013; Bleem et al. 2015; Planck Collaboration XXVII 2016), and to constrain cosmological parameters using largescale structure matter distribution (Hasselfield et al. 2013; Mantz et al. 2015; Planck Collaboration XXIV 2016; Planck Collaboration XXII 2016; de Haan et al. 2016). However, cosmological constraints obtained via the tSZ effect strongly depend on the massobservable relation. Assuming a value of b = 0.2 (Planck Collaboration XXIV 2016) for the hydrostatic mass bias, the observed number of galaxy clusters on the sky is only half of the predicted number assuming CMBderived cosmological parameters (Planck Collaboration XIII 2016). CMBderived cosmological parameter, favours a hydrostatic mass bias of b = 0.3–0.5. Thus, methodology to break the degeneracy between cosmological parameters and the massobservable relation needs to be investigated further to test the consistency of the ΛCDM cosmological model.
After the tSZ effect, the second dominant source of arcminutescale anisotropies is the kinematic SZ (kSZ) effect (Ostriker & Vishniac 1986), produced by the peculiar motion of galaxy clusters with respect to CMB restframe. The kSZ is approximately ten times fainter than the tSZ effect and contrary to the tSZ effect, it cannot be separated from the CMB signal using spectral considerations. Consequently, the measurement of this kSZ effect is significantly harder than the one of the tSZ effect. The kSZ effect is directly related to the peculiar velocity field of the matter distribution and the baryonic matter density. Thus, this effect presents significantly different cosmological dependancies than the tSZ effect, especially with the universe growth rate (Sugiyama et al. 2017), which is a powerful probe to understand the CMBLSS tension for cosmological parameter estimation.
Several approaches have been proposed to recover the kSZ effect: (i) Direct measurement of anisotropies in the CMB angular power spectrum at highℓ (see Addison et al. 2013; George et al. 2015, for recent results); (ii) the pairwise momentum estimator, using the preferential motion of one largescale structure toward another largescale structure (Peebles 1980; Diaferio et al. 2000); (iii) inverting the continuity equation relating density and velocity fields (Ho et al. 2009; Kitaura et al. 2012); and (iv) crosscorrelation bispectrum between CMB and cosmic shear (Doré et al. 2004) or galaxy surveys (DeDeo et al. 2005)
Evidence of the kSZ effect angular power spectrum has been found using the CMB angular power spectrum (George et al. 2015) and combining the tSZ angular powerspectrum and bispectrum (Crawford et al. 2014). The kSZ detection has been achieved in two cases: for individual galaxy cluster internal dynamics (Sayers et al. 2013; Adam et al. 2017), and for a statistical sample of galaxy clusters using pairwise momentum (Hand et al. 2012; HernándezMonteagudo et al. 2015; Soergel et al. 2016). Recent studies (see e.g., HernándezMonteagudo et al. 2015) achieve statistical detection at approximately 5σ significance level.
The kSZ effect produces a nonGaussian contribution to the CMB anisotropies. Highorder statistics have been shown as a powerful probe to detect the tSZ effect (see e.g., Wilson et al. 2012; Hill & Sherwin 2013; Planck Collaboration XXII 2016). In this paper, we explore the possibility to detect the tSZkSZkSZ crosscorrelation bispectrum for the next generation of CMB experiments. First, in Sect. 2 we present tSZ and kSZ effects, and in Sect. 3 we detail the computation of tSZ and kSZ power spectra. Then, in Sect. 4 we present the modeling of the tSZ and kSZ bispectra. Finally, in Sect. 5 we present our forecasts for next generation CMB experiments.
2. The SZ effects
The tSZ effect is a distortion of the CMB black body radiation through inverse Compton scattering. CMB photons receive an average energy boost by collision with hot (a few keV) ionized electrons of the intracluster medium (see e.g., Birkinshaw 1999; Carlstrom et al. 2002, for reviews). The thermal SZ Compton parameter in a given direction, n, on the sky is given by (1)where k_{B} is the Boltzmann constant, c the speed of light, m_{e} the electron mass, σ_{T} the Thomson crosssection, ds the distance along the lineofsight, n, and n_{e} and T_{e} are the electron number density and temperature, respectively. In units of CMB temperature the contribution of the tSZ effect for a given observation frequency ν is (2)where T_{CMB} is the CMB temperature, and g(ν) the tSZ spectral dependance. Neglecting relativistic corrections we have (3)with x = hν/ (k_{B}T_{CMB}). At z = 0, where T_{CMB}(z = 0) = 2.726 ± 0.001 K (Mather et al. 1999), the tSZ effect is negative below 217 GHz and positive for higher frequencies.
The kSZ effect produces a shift of the CMB black body radiation temperature but causes whiteout of spectral distortion. Contrary to tSZ effect, it thus possesses the same spectral energy distribution (SED) as the CMB primordial anisotropies and cannot be separated from them. The kinetic SZinduced temperature anisotropies in a given direction on the sky is given by (4)with v the peculiar velocity vector of the galaxy cluster. The total kSZ flux from a cluster is proportional to the gas mass, M_{gas,500} of the cluster, and modulated by the scalar product v·n that ranges from − v to v.
3. Power spectra
3.1. General formalism
The angular cross power spectrum between two maps reads (5)with x_{ℓm} and y_{ℓm}, the coefficients from the spherical harmonics decomposition of the two maps concerned, and ℓ the multipole of the spherical harmonic expansion. In the context of largescale structure tracers, we model this crosscorrelation, as well as the auto correlation power spectra, assuming the following general expression (6)where is the Poissonian contribution and is the twohalo term. These terms can be computed considering a mass function formalism. The mass function, d^{2}N/ dMdV, gives the number of dark matter halos (in this paper we consider the fitting formula from Tinker et al. 2008) as a function of the halo mass and redshift.
The Poissonian term can be computed by assuming the Fourier transform of normalized halo projected profiles of X and Y, weighted by the mass function and the respective fluxes of the halo for X and Y observables (see e.g., Cole & Kaiser 1988; Komatsu & Seljak 2002, for a derivation of the tSZ autocorrelation angular power spectrum). (7)where X_{500} and Y_{500} are the average fluxes of the halo in X and Y maps that depend on the critical mass of the galaxy cluster, M_{500}, the redshift, z, and can be obtained with scaling relations, and dV/ dzdΩ, the comoving volume element. The Fourier transform of a 3D profile projected across the lineofsight on the sphere reads, , where p(x) is the halo 3D profile in X or Y maps, x = r/r_{s}, ℓ_{s} = D_{ang}(z) /r_{s}, and r_{s} is the scale radius of the profile.
The twohalo term corresponds to largescale fluctuations of the dark matter field, that induce correlations in the halo distribution over the sky. It can be computed as (see e.g., Komatsu & Kitayama 1999; Diego & Majumdar 2004; Taburet et al. 2011) (8)with b(M_{500},z), the time dependent linear bias that relates the power spectrum between X and Y distribution, P_{X,Y}(k,z), to the underlying dark matter power spectrum. Following Mo & White (1996), Komatsu & Kitayama (1999) we adopt
with , D_{g}(z), the linear growth factor, and δ_{c}(z), the overdensity threshold for spherical collapse.
3.2. The tSZ and kSZ scaling relations
A key step in the modeling of the crosscorrelation between tSZ and kSZ is to relate the mass, M_{500}, and the redshift, z, of a given cluster to tSZ flux, Y_{500}, and kSZ flux, K_{500}. The crosscorrelation between tSZ and kSZ effects is thus highly dependent on the M_{500} − Y_{500} and the M_{500} − K_{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. However, such observational constraints are not available for the kSZ effect. 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.
Scalinglaw parameters and error budget for both Y_{500} − M_{500} (Planck Collaboration XXIX 2014) and Y_{500} − T_{500} (Planck Collaboration XXIX 2014) relations.
We used the M_{500} − Y_{500} scaling laws presented in Planck Collaboration XXIX (2014): (9)with h the dimensionless Hubble parameter, D_{A} the angular distance, and E(z) = Ω_{m}(1 + z)^{3} + Ω_{Λ}. The coefficients Y_{⋆}, α_{sz}, and β_{sz} from Planck Collaboration XXIX (2014), are given in Table 1. We used b = 0.2 for the bias between Xrayestimated mass and effective mass of the clusters.
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): (10)where the coefficients T_{⋆}, α_{T} , and β_{T} are given in Table 1.
To model the K_{500} − M_{500} relation, we consider the relation from DeDeo et al. (2005) for the velocity field: (11)and consequently for the velocity dispersion (12)with D_{g} the growth factor and σ_{j}(z) is defined for any integer j as (13)where is the Fourier transform of the real space tophat window function, with , where is the critical density of the universe.
By simplicity, to estimate the kSZ flux rootmeansquare (over velocities), K_{500}, we consider the relation (14)K_{500} is proportional to M_{gas,500}, thus the K_{500} − M_{500} and M_{gas,500} − M_{500} intrinsic scatters are identical. The extra scatter induced by the velocities is accounted by the σ_{v} factor. We derived a K_{500} − M_{500} intrinsic scatter of σ_{log K} = 0.03 ± 0.01 from the galaxy cluster sample presented in Planck Collaboration XI (2011).
Additionally, the Y_{500} − M_{500} and K_{500} − M_{500} intrinsic scatters are correlated. The intrinsic scatters follow the relation (15)where σ_{log YK} = 0.08 ± 0.01 is the intrinsic scatter of the Y_{500} − K_{500} (equivalent to the scatter of the Y_{500} − M_{gas,500}) and ρ is the correlation factor between Y_{500} − M_{500} and K_{500} − M_{500} intrinsic scatters. For consistency, we used σ_{log Y} = 0.10 ± 0.01 that has been derived on the galaxy cluster sample from Planck Collaboration XI (2011). We derived ρ ∈ [ 0.5,1.0 ]. In the following we assume ρ = 0.75.
3.3. Lognormal scatter and npoint correlation functions
Scaling relations shown in Sect. 3.2 presents an intrinsic physical scatter (see Table 1). This scatter is generally considered as a lognormal distribution. When entering in correlation functions, this lognormal scatter will act as an additional source of power. This additional power, for a quantity X, can be expressed as a relation between ⟨X^{n}⟩ and X_{⋆}, where X_{⋆} is the lognormal mean of the X variable. In the most general case, the nth momentum, M^{(n)}, expectation of a set of N variable X_{i} can be written (16)where A_{norm} is the normalization factor of the lognormal distribution, is the scaling relation scatter covariance matrix, X is a vector of X_{i} variables, X_{⋆} is a vector containing the lognormal expectation X_{⋆,i} for each variable X_{i}, and n_{i} is the order of each variable X_{i} in the momentum M^{(n)} and satisfies . It can be easily shown that (17)where n is a vector of n_{i}. This effect produces a power enhancement of 6% for the tSZ bispectrum, which is significant for highsignaltonoise ration (S/N) measurements of tSZ bispectrum. For the tSZkSZkSZ bispectrum, we derived a power enhancement of 1.8%. In the following, we correct all spectra for this effect.
3.4. 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 (18)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. 3.2.
To model the kSZ profile, we need the density, n_{e}(r), profile. 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 kSZ varies with n_{e}, the kSZ profile is proportional to P(r)^{ϵp}, where ranges from 0.5 to 1.0 for 1.0 <δ< ∞.
The overall normalization of kSZ profile is deduced from the scaling law K_{500} − M_{500} presented in Sect. 3.2.
Amplitude of the different terms’ contribution to the tSZ and kSZ power spectra and bispectra.
Fig. 1 Power density as a function of M_{500} (top panel) and redshift (bottom panel) for the tSZ power spectrum (dotted blue line), the tSZ bispectrum (solid blue line), the kSZ power spectrum (dotted red line), and the tSZkSZkSZ bispectrum (solid red line). 

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3.5. tSZ and kSZ power spectra
The kSZ effect is dependent on the orientation of the peculiar velocity vector. Consequently, power spectra have to be averaged over orientations. In Table 2, we present the multiplicative factors to be applied on kSZrelated power spectra, where P_{B} is the power spectrum of the momentum field and P_{m} is the power spectrum of the matter field. We notice that kSZ effects present a twohalo term, induced by the largescale correlations between the velocities of different clusters.
In Fig. 1, we present the power density as a function of M_{500} and z for the tSZ and kSZ power spectra, the tSZ bispectrum, and the tSZkSZkSZ crossbispectrum. We observe that the tSZ power spectrum samples higher mass and lower redshift than the kSZ power spectrum; this effect is a consequence of the slope of Y_{500} − M_{500} (1.7) and K_{500} − M_{500} (1). We also observe that bispectra give more weight to very massive and nearby objects.
4. Bispectra
Combining tSZ and kSZ, it is possible to build two autocorrelation bispectra, and two crosscorrelation bispectra. In this work we aim at predicting the tSZkSZkSZ bispectrum, thus our modeling only accounts for nonGaussian objects that correlate with the tSZ effect. Consequently, we only consider the kSZ effect from virialized halos and we neglect the kSZ contribution produced by diffuse baryons at higher redshift.
4.1. Crosscorrelation bispectra
Following the same halo model approach, we can easily predict the SZ bispectra (see Bhattacharya et al. 2012, for a detailed description of the tSZ bispectrum). In halo model formalism, a bispectrum can be separated into three terms: onehalo, twohalo, and threehalo, as (19)The onehalo term, is produce by the autocorrelation of a cluster with itself, (20)The twohalo involves two points from the same halo and a third from another one. As a consequence, this term receives three contributions, (21)The threehalo involves the correlation of three different halos, (22)with B_{X,Y,Z}(k_{1},k_{2},k_{3},z) being the bispectrum of X, Y, and Z distribution, and b_{3}(M_{500},z) being the bias that relates darkmatter and halo distributions.
4.2. The tSZ and kSZ crossbispectrum
Fig. 2 tSZ (top panel) and tSZkSZkSZ (bottom panel) bispectra 1halo term (blue line), 2halo term (red line), and 3halo term (green line). 

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The tSZtSZkSZ bispectrum has a null expectation due to the average over all the directions for the momentum field of galaxy clusters. Table 2 lists the relative amplitude of each contribution.
For the prediction of the tSZkSZkSZ twohalo term, we compute the momentum field power spectrum similarly to Shaw et al. (2012). The tSZkSZkSZ bispectrum twohalo term involves three contributions; two of which where the two considered halos receive contribution from the kSZ effect. These terms involve the momentum field power spectrum. The third contribution involves the correlation between a halo weighted by its tSZ flux and a halo weighed by the square of the kSZ effect. Consequently, this third contribution involves the matter power spectrum that describes the distribution of halos over the sky, weighted by the average over momentum directions (similarly to the kSZ power spectrum). For the 3halo term mattermomentum bispectrum we used the prescription from (DeDeo et al. 2005).
We studied the cosmological parameter dependancies of this cross bispectrum. We found that the tSZkSZkSZ bispectrum is proportional to . By comparison the tSZ bispectrum is proportional to and ℓ ≃ 1000. Then, tSZkSZkSZ might be a powerful probe to break degeneracies between cosmological parameters and scaling relations by providing similar cosmological dependencies to the tSZ power spectrum with significantly different astrophysical process dependencies. We also note that the tSZ bispectrum presents similar degeneracies to the tSZ power spectrum. Consequently, it cannot be used to break degeneracies between cosmology and astrophysical processes.
In Fig. 1, we present the power density for tSZ and tSZkSZkSZ bispectra. We observe that bispectra are sensitive to higher mass and lower redshift than power spectra. We note that the tSZkSZkSZ bispectrum and tSZ power spectrum present similar power density distribution as a function of mass and redshift. The tSZkSZkSZ bispectrum is thus sensitive to galaxy clusters with mass ranging from 10^{14} and 10^{15}M_{⊙}, at z< 1.
We present the tSZ and tSZkSZkSZ bispectra in Fig. 2. We observed that for the tSZ bispectrum, the onehalo term is two orders of magnitude higher than the twohalo term. Consequently, for the tSZ bispectrum, twohalo and threehalo terms can be safely neglected. This is consistent with the tSZ angular power spectrum that only presents significant contribution from the twohalo term at very lowℓ. When using higherorder statistics we favor higher mass, lowerz objects that are less frequent over the sky. Indeed, the onehalo term amplitude evolved with the number, N_{cl}, that significantly contributes to the spectra, where the twohalo term evolves as and the threehalo term as .
Contrary to the tSZ bispectrum, the tSZkSZkSZ bispectrum twohalo and threehalo terms present significant contributions compared to the onehalo term. This higher contribution is explained by the fact that the kSZ effect favors lowermass and higherz objects than the tSZ effect, as shown in Fig. 1.
4.3. Uncertainties and optimal estimator
To compute the bispectrum between three maps X, Y, and Z, we considered the following formula (23)where X_{ℓ1}, Y_{ℓ2}, and Z_{ℓ3} are the real space map that only contain ℓ_{1}, ℓ_{2}, ℓ_{3} multipoles of maps X, Y, Z respectively, f_{sky} is the covered sky fraction, and n is the direction over the sky.
This estimator is known to reduce the variance of the bispectrum without biasing the expectation of the bispectrum. We note that, in the case of a crosscorrelation bispectrum ℓ_{1}, ℓ_{2}, ℓ_{3} are not commutative quantities.
Then the bispectrum variance in the weak nonGaussian limit can be expressed as (24)with N_{ℓ1ℓ2ℓ3}, being the number of modes for the (ℓ_{1},ℓ_{2},ℓ_{3}) triangle.
For our purpose, we considered the tSZkSZkSZ bispectrum, thus we have two points that are identical. Considering that the expectation of tSZ and kSZ crosscorrelation power spectrum, , is zero, we can safely neglect this term in the computation of uncertainties. In this context Eq. (24)reduces to (25)If we have ℓ_{1} = ℓ_{2} then (26)and if we have ℓ_{1} ≠ ℓ_{2} then (27)
4.4. Foreground contamination
The two main foreground contaminations in SZ analyses are the CIB and extragalactic radio sources.
CIB leakage in tSZ and CMB maps is composed of highz CIB sources. As a consequence, CIB in such maps is almost Gaussian and will not significantly bias the results. The level of CIB in the final bispectrum can be estimated through the different shape of CIB and SZ bispectra Lacasa (2014).
Radio sources with a significant flux are in small number on the microwave sky. It has been shown in tSZ bispectra analyses (Planck Collaboration XXI 2014) that the results are not biased by radio sources that would produce a negative bias due to the way radio sources leak in tSZ maps (Hurier et al. 2013).
5. Forecasts
5.1. Cosmicvariancelimited experiment
In a first step, we estimated the expected signaltonoise ratio considering a sky composed by tSZ, kSZ, and CMB anisotropies. Indeed tSZ can be extracted from other components (Hurier et al. 2013), but the kSZ signal cannot be distinguished from the CMB primary anisotropies or secondary anisotropies that follow the CMB black body emission law. We also consider f_{sky} = 0.5. This choice of a small sky fraction is used to avoid foreground residual contamination that contaminates tSZ and CMB signals.
We present in Fig. 3 the expected cumulative S/N as a function of the multipole ℓ. We observed that the tSZkSZkSZ can be detected for scales above ℓ ≃ 3000.
5.2. Realistic CMB experiments
The Planck experiments have produced a cosmicvariance limited measurement of the CMB temperature angular power spectrum (Planck Collaboration XV 2014). However, secondary anisotropies (such as the tSZ effect) measurements are still dominated by instrumental noise. In this section we present the expected S/N from the Planck experiment and from a future COrE+^{1} like CMB mission.
Fig. 3 Cumulative S/N for the tSZkSZkSZ bispectrum as a function of ℓ for a cosmicvariancelimited experiment (black line), the Planck experiment (blue line), and a COrE+ like experiment (red line). The dashed line shows the 5σ level. 

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5.2.1. Plancklike experiment
To estimate the expected S/N using data from a Plancklike mission, we used the noise level from Planck public CMB (Planck Collaboration IX 2016) and tSZ (Planck Collaboration XXII 2016) maps assuming f_{sky} = 0.5. For both CMB and tSZ maps we used the total measured powerspectra to estimate the noise level in the tSZkSZkSZ bispectrum. Consequently, this estimation of the noise level accounts for CIB and point source contamination in both tSZ and CMB maps.
In Fig. 3, we present the expected signaltonoise as a function of the maximum ℓ considered. We observe that Planck is expected to achieve a 0.2σ measurement of the tSZkSZkSZ bispectrum. This rules out the possibility of extracting this signal from the Planck data. The main limitations are the noise level in tSZ and CMB maps, residual from other astrophysical components (mainly the CIB for the tSZ map), and the angular resolution of Planck maps, 5 arcmin FWHM for the CMB maps and 10 arcmin for the tSZ maps. The CMB also strongly limits the sensitivity at lowℓ. In Fig. 3, we can see lowℓ oscillations that correspond to the CMB acoustic oscillation pics contributing to the noise for the tSZkSZkSZ bispectrum.
Fig. 4 Likelihood for combined tSZ power spectrum, bispectrum, and tSZkSZkSZ bispectrum as a function of Y_{⋆}, σ_{8}, and Ω_{m}. Dark blue, blue, and light blue contours indicate the 1, 2, and 3σ levels. 

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5.2.2. Future CMB experiments
We now consider a future CMB spacecraft mission assuming specificities (frequencies, noise level per detector, number of detectors, beams) for a COrE+ like experiment, and a sky coverage f_{sky} = 0.5. For such a future experiment, we do not have direct access to CMB or tSZ map noise levels. Thus, we computed the expected noise level in componentseparated maps obtained through linear combination of multifrequency intensity maps. The optimal noise level, V_{i}, for a single astrophysical component is given by, (28)where F_{i} is the component spectral behavior in the frequency channels of the experiment, and is the instrumental noise covariance matrix. We assume that is diagonal, and that multiple detectors at a given frequency have uncorrelated noise.
However, there are several components on the sky. Thus, we use the following equation (29)where ℱ is a rectangular matrix containing the astrophysical component spectral behavior. In this analysis, we considered:

the tSZ effect;

the CMB;

one thermal dust component, that follows a modified blackbody SED (SED) with a temperature, T_{d} = 20 K, and spectral index β_{d} = 1.6;

one radio component following a ν^{αr} SED with a spectral index α_{r} = − 1;

the CO component;

the spinning dust component.
Additionally, some components, such as the cosmicinfraredbackground (CIB), cannot be modeled with a single spectral law and contribute as a partially correlated component from frequency to frequency. We thus model the CIB contribution to the final variance as (30)where is the CIB covariance matrix. We computed the CIB covariance matrix using the model presented in (Planck Collaboration XXIII 2016). For a more realistic estimation we performed the noise estimation as a function of the multipole ℓ to have an estimate of the noise powerspectrum in CMB and tSZ maps. We verified that this approach is realistic by applying it to the Planck mission specificities and comparing to the noise level observed in Planck tSZ and CMB public maps.
In Fig. 3, we present the expected tSZkSZkSZ bispectrum S/N for a COrE+ like experiment. We observe that the tSZkSZkSZ S/N is dominated by the cosmic variance up to ℓ = 4000. We can expect a detection up to 200σ when neglecting potential systematic effects in the CMB and tSZ maps. We also stress that our estimation of the tSZkSZkSZ bispectrum may underestimate the real signal at highℓ, where the internal dynamics of the gas inside galaxy clusters will most likely add extra power on small scales.
In Fig. 4, we present the expected constraints on Ω_{m}, σ_{8}, and Y_{⋆} when combining the measurement of tSZ power spectrum, bispectrum, and tSZkSZkSZ bispectrum.
The Likelihood function is computed as follows, (31)where D is a vector containing the tSZtSZtSZ and tSZkSZkSZ bispectra for our fiducial model (σ_{8} = 0.8, Ω_{m} = 0.3, ΔY_{⋆}/Y_{⋆} = 0), M is a vector containing the two bispectra for parameters (σ_{8}, Ω_{m}, ΔY_{⋆}/Y_{⋆}), and is the covariance matrix of the two bispectra in the weakly nonGaussian limit (see Eq. (24)). The correlation between the tSZtSZtSZ and tSZkSZkSZ bispectra in the weakly nonGaussian limit is proportional to , thus contraints from the tSZtSZSZ bispectrum and from the tSZkSZkSZ bispectrum can be considered as independent. Figure 4 shows the expected constraints when σ_{8}, Ω_{m}, and Y_{⋆} are allowed to vary. We marginalized over H_{0} = 67.8 ± 0.9 km s^{1} Mpc^{1} and fixed all other parameters.
We observed that σ_{8} and Ω_{m} present a high degree of degeneracy, as our three probes present similar degeneracies for these two parameters. However, we observed that we can achieve a precision of 4% on Y_{⋆} without any external prior. Such an approach will allow us to calibrate the tSZ scaling relation without the need for Xray hydrostatic mass.
6. Conclusion and discussion
We have proposed a new method to detect kSZ effect using future highresolution CMB experiments. This method presents the advantage of being sensitive to the galaxy cluster velocity dispersion without bias from CMB autocorrelation or from kSZ effect produced by the diffuse baryonic gas at high redshift. This method also allows to constrain the velocity field without a selection function. By comparison, the kSZ angular power spectrum measured by George et al. (2015), is sensitive to the total kSZ power spectrum. The constraints achieved on the kSZ power spectrum by Crawford et al. (2014) when combining tSZ power spectrum and bispectrum is an indirect constraint, and is affected by a strong cosmic variance, as it measures the tSZtSZtSZ and tSZkSZkSZ bispectra as a single quantity.
The method proposed in the present paper, relies on direct measurement of the tSZkSZkSZ bispectrum after a separation of the tSZ and kSZ signals. Consequently, it allows us to obtain a modelindependent and lowcosmicvariance estimation of the tSZkSZkSZ bispectrum.
We have presented a complete modeling of the tSZkSZkSZ bispectrum and deduced the associated cosmological parameter dependencies. We derived the dependencies of the tSZkSZkSZ and tSZ bispectra with respect to cosmological parameters. Previous works have also discussed the tSZ bispectrum scaling with cosmological parameters (see e.g., Bhattacharya et al. 2012; Crawford et al. 2014) and found slightly different scaling. The cosmological dependencies of the tSZ angular power spectrum are highly dependent on the weighting given to each halo. As a consequence, differences on the massobservable relation will significantly affect the bispectrum scaling with respect to cosmological parameters. This also explains the scale dependance of the bispectrum scalings. Indeed, different angular scales receive contribution from halos at differeérent redshifts and masses.
We also demonstrated that future experiments, will be sensitive to the tSZkSZkSZ crosscorrelation bispectrum up to 200σ.
We demonstrated that the tSZkSZkSZ bispectrum can be combined with the tSZ power spectrum and bispectrum to set tight constraints (4%) on the Y − M relation calibration and thus on the hydrostatic mass bias. This will enable the possibility of setting cosmological constraints without the need for prior on the hydrostatic mass bias, which is a crucial step considering that the hydrostatic massbias is the main limitation for cosmological parameter estimation from tSZ surveys.
Acknowledgments
The author thanks F.Lacasa for useful discussions. We acknowledge the support of the French “Agence Nationale de la Recherche” under grant ANR11BD56015.
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All Tables
Scalinglaw parameters and error budget for both Y_{500} − M_{500} (Planck Collaboration XXIX 2014) and Y_{500} − T_{500} (Planck Collaboration XXIX 2014) relations.
Amplitude of the different terms’ contribution to the tSZ and kSZ power spectra and bispectra.
All Figures
Fig. 1 Power density as a function of M_{500} (top panel) and redshift (bottom panel) for the tSZ power spectrum (dotted blue line), the tSZ bispectrum (solid blue line), the kSZ power spectrum (dotted red line), and the tSZkSZkSZ bispectrum (solid red line). 

Open with DEXTER  
In the text 
Fig. 2 tSZ (top panel) and tSZkSZkSZ (bottom panel) bispectra 1halo term (blue line), 2halo term (red line), and 3halo term (green line). 

Open with DEXTER  
In the text 
Fig. 3 Cumulative S/N for the tSZkSZkSZ bispectrum as a function of ℓ for a cosmicvariancelimited experiment (black line), the Planck experiment (blue line), and a COrE+ like experiment (red line). The dashed line shows the 5σ level. 

Open with DEXTER  
In the text 
Fig. 4 Likelihood for combined tSZ power spectrum, bispectrum, and tSZkSZkSZ bispectrum as a function of Y_{⋆}, σ_{8}, and Ω_{m}. Dark blue, blue, and light blue contours indicate the 1, 2, and 3σ levels. 

Open with DEXTER  
In the text 