Dark Energy from the Thermal Sunyaev Zeldovich Power Spectrum

Transcription

1 Dark Energy from the Therma Sunyaev Zedovich Power Spectrum Boris Boiet, 1, 2, 3 Barbara Comis, 2 Eiichiro Komatsu, 4 and Juan Macias Perez 2 1 Écoe Normae Supérieure de Lyon, 46 Aée d Itaie, Lyon 69007, France 2 Laboratoire de Physique Subatomique et de Cosmoogie, Université Grenobe-Apes, CNRS/IN2P3, 53, avenue des Martyrs, Grenobe cedex, France 3 Jodre Bank Centre for Astrophysics, Schoo of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, U.K. 4 Max-Panck-Institut für Astrophysik, Garching bei München, Germany The work presented in this paper is dedicated to the cosmoogica constraints that can be obtained from the therma Sunyaev Zedovich (tsz) effect anguar power spectrum. The numerica computations of the tsz power spectrum is carried out with a new modue added to the Botzmann code CLASS which aows for a straightforward interfacing with the Markov Chain Monte Caro (MCMC) samper Montepython. We revisit the Panck Coaboration constraints on the parameter combination 8 Ω3, and show that the trispectrum contribution to the covariance matrix can not m be negected for the measured mutipoes 1000, in agreement with a recent anaysis by Horowitz and Sejak. Unike in the Panck anaysis, the whoe set of cosmoogica parameters are samped in our MCMC anaysis, aong with the mass bias B (1 b) 1. The ampitude of the modeed tsz power spectrum is found to scae with the parameter combination: F (Ω m /B) 3/8 h 1/5. We present the constraints on the F -parameter from the Panck 2015 Compton-y parameter power spectrum data: F = ± Interestingy, our best-fit tsz power spectrum extrapoated at = 3000 matches perfecty the / data. To accommodate the SZ constraint on F with Panck 2015 Cosmic Microwave Background (CMB) temperature anisotropy data, we find that the mass bias has to be set to B = 1.61 ± 0.16, i.e. (1 b) = 0.62 ± In the ast part of this anaysis, the dark energy equation of state is set as a free parameter, w. Athough the current tsz data does not aow for the determination of w, we find that adding a prior on the Hubbe parameter h = 0.72 ± 0.03, corresponding to the atest oca measurements, enabes a constraint on w that is competitive with the one obtained from the CMB ensing data. Our resuts for the wcdm cosmoogy are: w = 1.13 ± 0.14, = ± and Ω m = ± 0.025, consistent with CMB constraints. I. INTRODUCTION The Sunyaev Zedovich (SZ) effect [1] is a frequency dependent distortion of the cosmic microwave background (CMB) due to the inverse Compton scattering of the CMB photons off the eectrons in the intra-custer medium (ICM). The ICM is assumed to be a diffuse pasma at hydrostatic equiibrium within the dark matter potentia wes. There are two main components to the SZ effect. One that is due to the interna motion of the eectrons, the so-caed therma SZ effect (tsz), and one caused by the buk motion of the gas when the custer is moving with respect to the CMB rest-frame, the so-caed kinetic SZ effect (ksz). The contribution from the ksz effect is at east one order of magnitude smaer than the tsz effect, when we compare the anguar anisotropy power spectra of the CMB temperature anisotropy associated with each component. Therefore the ksz contribution is negected in this work. If we assume the eectrons in the ICM gas to be nonreativistic, the frequency dependence of the tsz effect is described by the spectra function g ν (x) = x coth (x/2) 4, with x = hν k B T CMB. (1) This means that the temperature of the CMB photons that transit across a gaaxy custer is shifted by T T CMB = g ν y, (2) where the Compton-y parameter, which sets the ampitude of the effect, contains the information reative to the therma properties of the ICM y ds n ek B T e m e c 2 σ T = ds P e m e c 2 σ T, (3) where σ T is the Thomson cross-section, n e and P e are the eectron number density and pressure, respectivey, and ds is the ine eement aong the ine of sight. The most recent measurements of the a-sky y-map were performed by the Panck Coaboration in Reca that the main products of the Panck mission are temperature maps of the CMB at anguar resoutions from 33 arcmin to 5 arcmin, with nine frequency bands centered at 30, 44, 70, 100, 143, 217, 353, 545 and 857 GHz. So, based on the tsz spectra signature (1), it is possibe to extract two types of information from the Panck maps. First, one can appy a series of agorithms [2] to identify individua custers. The Panck coaboration reported severa hundreds of resoved custers (RC) up to redshift z 1 and within the mass range M. The mass of a candidate custer is not directy accessibe. It is

2 2 deduced from the scaing reation between the integrated Compton-y contribution over the soid ange spanned by the custer, Y ydω (4) and the over-density mass M 500, with respect to the critica density of the universe. By negecting a contributions due to non-gravitationa processes, assuming spherica coapse for the dark matter hao and hydrostatic equiibrium of the ICM, a simpe mode to reate the physica parameters of custers can be buit [3]. The integrated Compton parameter Y can be reated to the custer tota mass M through a simpe power aw. The Y M reation can then be tested against data and simuations [4]. However, not ony hydrostatica equiibrium may not be reached, due to non-therma pressure in the hao [5], there may aso be issues reated with systematics of the X-ray observations that can affect the mass estimate. To account for these facts, a free parameter, the so-caed mass bias B, is introduced in order to transate M 500 into the true hao mass. As we sha see, the mass bias pays a crucia roe for the cosmoogica parameter extraction from the tsz effect. Second, apart from the custer cataogue, the wide sky coverage and the mutipe frequency bands of the Panck mission aows to compute the a-sky Compton parameter y-map. This is done using the Modified Interna Linear Combination Agorithm (MILCA) and Needet Independent Linear Combination (NILC) methods for components separation [6, 7]. Then, the statistica properties of the Compton y parameter can be anayzed in harmonic space, in a way simiar to the CMB temperature maps. The main observabe we consider here is the tow-points correator that yieds the anguar anisotropy power spectra of the y-maps in harmonic space, denoted C y2. Note that even ow mass or not individuay resoved custers aso contribute to this observabe, athough they can not be detected individuay. Higher oder correators have aso been considered in recent works [8, 9], but we chose to not discuss them, as our main focus here is to identify the best constraints that can be set on our the cosmic evoution based on the tsz power spectrum aone. To estimate the anguar power spectrum of the y-maps it is necessary to correct for the beam convoution as we as statistica noise due to mode-couping induced by masking foreground-contaminated sky regions. For the SZ data, this is done by expoiting the XSPECT method [10] that uses cross-power spectra between different y- maps. The fina power spectrum obtained with this procedure does not contain excusivey the contribution from the tsz effect associated with gaaxy custers, C tsz. It aso contains a contribution from severa foregrounds: notaby, the cosmic infrared background (CIB), radio sources (RS) and infrared point sources (IR). Moreover, on sma anguar scaes ( 1400) the y 2 power spectrum is dominated by correated noise (CN). The frequency dependence of the power spectra for these foregrounds can be accuratey estimated using the Fu Foca Pane (FFP6) simuations [11] and physicay motivated modes [12]. Nevertheess, their normaisations remain undetermined and have to be treated as free parameters. Therefore, for the measured power spectrum Ĉy2, we assume a five components mode: C y2 = C tsz + A CIB C CIB + A IR C IR + A RS C RS + A CN C CN, (5) where C CIB, C IR, C RS, C CN are the foreground contaminants, tabuated and reported in tabe V. (A hat refers to the measured signa and no hat refer to the mode.) Since correated noise argey dominates over other foregrounds and the tsz component at high mutipoes, its ampitude can be set by fitting the signa at = 2742: A CN = Ĉy2 CN 2742 /C2742 = 0.903, (6) as can be computed with the numerica vaues in the ast ine of tabe (V). This eaves three undetermined foreground parameters, namey A CIB, A IR and A RS. A crucia information can be extracted from the cataogue of confirmed custers detected via the SZ effect: the power spectrum of the combined foregrounds can not be arger than the difference between the measured tota power spectrum, Ĉ y2, and the power spectrum associated with the projection of the SZ signa from resoved custers on the y-map, Ĉ RC. If we negect systematic and statistica uncertainties, this means that the foreground ampitudes have to satisfy the foowing inequaity: A CIB C CIB +A IR C IR +A RS C RS +A CN C CN < Ĉ y2 Ĉ RC. (7) This condition is impemented in the maximum ikeihood anaysis for the determination of cosmoogica parameters based on the Panck 2015 SZ data, presented beow. The tsz power spectrum C tsz is modeed anayticay, foowing the hao mode of Komatsu and Sejak [13]. It is sensitive to the cosmoogica parameters, in particuar: the reduced Hubbe constant, h = (H 0 /100) ; the ampitude of custering, ; the matter density parameter Ω m = 1 Ω de ; and the equation of state parameter of dark energy, w de. In the next section, the main steps of the cacuation of the tsz power spectrum and its numerica impementation are recaed. This incudes a discussion on the use of the concentration mass reations for the viria mass to overdensity mass conversion. In section III, we present the settings of our MCMC anaysis and we show the importance of the non-gaussian contribution to the covariance matrix. Our resuts regarding cosmoogica constraints are given in section IV: we start by revisiting the Panck 2015 anaysis, then we incude the trispectrum in

3 3 the covariance matrix and finay we perform a MCMC samping were the mass bias as we as the six cosmoogica parameters are varying. In the ast part of section IV, we expore the constraining power of the tsz data with respect to the equation of state parameter of dark energy. In section V, we summarise our main resuts and concusions. II. ANALYTICAL MODEL FOR THE THERMAL SUNYAEV ZELDOVICH POWER SPECTRUM In this section we describe our mode for the computation of the tsz anguar power spectrum. The main ingredients are the hao mass function (HMF), the pressure profie of the ICM and the inear matter power spectrum. We consider ony the 1-hao contribution, as the 2-hao term contribution to the tsz power spectrum is not significant, given the precision of the current data, see e.g. [14]. For the numerica cacuations of the tsz power spectrum (and trispectrum), we have deveoped a version of cass augmented with a tsz modue. The code is dubbed cass_sz and is avaiabe on the internet 1. Our code agrees with the fortran code szfast [15] within one percent for the the computation of the tsz power spectrum. The tsz anguar power spectrum is cacuated via an integration over mass and redshift of the two dimensiona Fourier transform of the hao pressure profie, zmax C tsz = dz dv z min dz n Mmax n M min d n M dn d n M y (M, z) 2, (8) where V is the comoving voume of the universe per steradian. Its derivative with respect to redshift is expressed as dv dz = (1 + z)2 d A (z) 2 ch 0 /H (z). (9) In our code, the integration over redshift is carried out with a simpe trapezoida rue from z min = 0 and up to z max = 6. At higher redshift the number density of haos is vanishingy sma. In fact setting z max = 3 as in [16] aso gives accurate resuts. The integration over the mass is performed using a Gaussian quadrature method within the mass range determined by M min = h 1 M and M max = h 1 M. (10) 1 website: The two dimensiona Fourier transform of the hao pressure profie reads as y = σ T m e c 2 4πr dxx 2 sin (x/ 500) x/ 500 P e (x), (11) where x r/r 500 with r the radia distance to the center of the hao, r 500 the radius of the sphere containing the overdensity mass M 500c with respect to the critica density of the universe, and 500 d A /r 500 with d A (z) the physica anguar diameter distance at redshift z. For the pressure profie, we use a standard generaized Navarro- Frenk-White (NFW) functiona: P e (x) = C P 0 (c 500 x) γ [1 + (c 500 x) α ] (γ β)/α, (12) where C is a dimension-fu quantity that depends on the over-density mass M 500c (via the Y M scaing reation). It is given by [ ] 2/ C = 1.65h 2 (H/H 70 0) 8/3 h70 M 500c ev cm 3. M (13) The reader is referred to Appendix D of [17] for further detais regarding this parametrization. In our main anaysis, the vaues of {γ, α, β, P 0, c 500 } are set to their best-fit vaues obtained within the Panck 2013 (P13) anaysis of custers X-ray data [18]. In cass_sz, the best-fit pressure profie of Arnaud et a 2010 (A10) [19], based on XMM Newton data set, is aso readiy avaiabe. The user may aso specify custom vaues of the NFW parameters. To account for the differences in mass estimates from weak-ensing and X- rays observation, we use a mass bias B to rescae the over-density mass M 500c used in the Fourier transform if the pressure profie as M 500c := M 500c /B. Note that this rescaing not ony affects the normaization of the pressure profie but aso its scae dependence via 500. Numericay, the integra in (12) is performed with Romberg s method between x = 10 6 and x = 10. To speed up our MCMC anaysis, since we were not interested in varying the NFW parameters in this work, we have tabuated the Fourier transform of the P13 and A10 pressure profies. The python code for the interpoation, and the tabuations, can be found in the subdirectory sz_auxiiary_fies of cass_sz. The differentia number density of haos in Eq. (8) depends on both mass and redshift and is written as dn d n M = 1 2 f (σ) ρ m (z min) d n σ 2 M d n M (14) where σ 2 is the variance of the matter over-density fied over a sphere of radius R (M, z) [3M/4πρ m (z)] 1/3, i.e. σ 2 (M, z) 0 dk k k 3 2π 2 P (k, z) W (kr)2 (15)

4 4 where P (k) is the inear matter power spectrum, W is the top-hat window function and f (σ) is the so-caed hao mass function (HMF) whose parametrisation can be deduced from N-body simuations. To simpify the task of accounting for the redshift dependency, σ 2 (M, z) can be factorized as σ 2 (M, z) = σ 2 (M, z min ) [D (z min ) /D (z)] 2 (16) where D (z) is the inear growth rate for matter perturbations. Then, instead of σ 2 (M, z), one can introduce an auxiiary variabe, ν (M, z), defined via ν (M, z) = δcrit/σ 2 2 (M, z), where δ crit = is the critica overdensity for spherica coapse [20]. Since the HMF, now a function of ν, is generay parametrized in terms of the over-density mass M X, we appy a chain rue: dn d n M = d n M X d n M d n M X d n M dn (M X, z) d n M X 1 d n σ 2 8πR 3 d n R f (ν) (17) with R [3M X /4πρ m (z min )] 1/3, where ν := ν (M X, z) and where we assumed d n M X /d n M 1 for the second equaity [15]. In fact, to write dn/d n M in this manner, an interpoation of σ 2 (M X, z min ) = σ 2 (R) is needed for a masses M X between M min and M max, so one can deduce ν and subsequenty evauate the HMF. A simiar interpoation for d n σ 2 /d n R is required. In our code, to obtain the interpoating functions for σ 2 and its derivative, given the inear matter power spectrum P (k) computed at z = z min, we evauate σ 2 (R) at various radii R between R min = h 1 Mpc (M M ) and R max = 54.9h 1 Mpc (M M ). At each R within this range, the integra (15) is performed with Romberg s method. Eventuay, the interpoation for σ 2 and its derivative with respect to R is obtained in terms of Chebyshev poynomias. In cass_sz, we have impemented four different parameterisations of the HMF: (i) the Bocquet et a (B15) caibration obtained with the Magneticum simuations, that captures the impact of baryons [21], as our main mode; (ii) the Tinker et a (T08) parametrization which is based on GADGET2 simuations [22]; (iii) the Tinker et a (T10) caibration, an updated version of T08 [20] ; (iv) the Jenkins et a (J01) parameterisation [23]. For instance, B15 or T08 HMF s are expressed as [ (σ ) ] a ( f (σ) = A + 1 exp c ) b σ 2, (18) with σ = δ crit / ν. The phenomenoogica parameters entering the HMF s are redshift dependent. The redshift dependency is generay parameterized via p = p 0 (1 + z) pz for a parameter p = A, a, b, c. Like we saw in formua (17), the HMF is specified in terms of the over-density mass M X. This coud be e.g. M 200m (over-density with respect to the mean matter density), M 180m as for the J01 HMF, or M 500c (over-density with respect to the critica density). Different strategies, corresponding to different choices of M X for the the HMF have been empoyed for the cacuation of the tsz power spectrum: The Panck Coaboration [8], as we as the authors of [9, 16], have chosen to perform the integra in (8) with the mass M being understood as M 500c. Then, no conversion is needed to compute the pressure profie which takes M 500c as an input. The HMF chosen by the authors of the aforementioned papers is aways T08. In the Tinker et a artice [22] (and in the reference for the updated T10 HMF too), the parameters of the HMF are tabuated for different over-density masses. Hence, by an interpoation, it is possibe to deduce the HMF parameters vaues corresponding to M 500c. This procedure is one of the impementations avaiabe in cass_sz, and we refer to the code for further detais on the conversion and interpoation. In szfast based artices, such as [14, 15, 21], the mass being integrated over is the viria mass M vir. A conversion from the viria mass to the overdensity mass is needed twice: for the HMF and for the pressure profie. The conversion is carried out using the Concentration-Mass (CM) reation for dark matter haos, such as the one obtained in Duffy et a 2008 (D08) [24]. In cass_sz, we have impemented severa CM reations, incuding D08, Kypin et a 2010 (K10) [25], Sanchez-Conde et a 2014 (SC14) [26] and Zhao et a 2009 (Z09) [27]. Again, for further detais regarding the conversion method the reader is refered to our code. These two strategies have ed to different concusions regarding the cosmoogica parameter constraints obtained from the SZ power spectrum data. In particuar, szfast based anaysis have found a higher vaue of than the Panck Coaboration, by about two standard deviations. Here we show that this discrepancy is not due to the use of different HMF as caimed in [21] but, in fact, to the ambiguity of the use of the CM reation. On figure 1 we present the tsz power spectrum computed in different setting. First, we observe on the topeft pane that the choice of the pressure profie is of no importance for < Second, on the top-right pane we see that the choice of the HMF does affect sighty the ampitude of the power spectrum. It is true that B15 yieds a smaer power spectrum than T08 or T10 when the HMF are evauated at M 200m, but this difference is

5 5 too sma to expain the discrepancy on the constraints obtained by the Panck Coaboration and szfast based anaysis. The bottom panes correspond to the tsz power spectrum computed using T08 HMF at the over-density mass M 1600m (eft pane) and M 200m (right pane) for severa CM reations. This means that the mass integration is performed over the viria mass, converted to the overdensity mass M 200m or M 1600m for the HMF and to M 500c for the pressure profie, via the CM reations. When the HMF is specified at M 1600m, a the resuts are consistent for the mutipoe range of interest < This is expected because with Ω m 0.3, one has M 1600m M 500c, so that the mass that is used for the pressure profie is actuay amost the same as the mass used in the HMF. But when the HMF is specified at M 200m, a the resuts disagree. This time, the ampitude of the discrepancy is significant and sha affect the constraints on cosmoogica parameters. In particuar, we see that the D08 CM reation eads to an underestimation of the tsz power spectrum compared to the spectrum obtained with T08 HMF computed at M 500c (and no CM reation is used) which corresponds to the soid back ine on both panes. Hence, to fit the Panck SZ data the use of D08 CM reation requires a arger tsz ampitude (arger ) compared to the method used by the Panck Coaboration. The fact that the CM reations disagree at M 200m is certainy due to the different cosmoogica settings of the N-body simuations which were empoyed to buid these reations. The szfast method (using viria mass to overdensity mass conversion) woud be consistent if one woud cacuate a new CM reation for each particuar setting of the cosmoogica parameters. In principe, this coud be done with the mandc code provided by Zhao et a in [27]. For our anaysis, we have preferred to adopt the same strategy as the Panck Coaboration and avoid the ambiguity reated to the CM reation. We sha now investigate the infuence of the cosmoogica parameters and the mass bias on the tsz power spectrum. This enabes us to determine the parameter combination that is best measured from the y-map data. The effects of the cosmoogica parameters and the mass bias on the tsz power spectrum are iustrated in figure 2. The optica depth at reionization, τ reio, has no impact on the ampitude of the tsz power spectrum (ess than one percent reative variation for 0.04 τ reio 0.12) and therefore it is not shown on the figure. For the mutipoe range of interest ( < 10 3 ), the equation of state (EoS) of dark energy w (bottom eft pane) and the spectra index n s (bottom right pane) have a minor effect on the ampitude of the tsz power spectrum. When one ooks at higher mutipoes, the spectra index shifts the ocation of the peak of the power spectrum (towards smaer anguar scaes for arger n s ) and the EoS changes the ampitude of the peak (increase of power for arger w). The mass bias B, the Hubbe parameter h, the ampitude of custering, and the matter density parameter Ω m are the determining parameters for the ampitude of the tsz power spectrum. We find that the scaing of the tsz power spectrum is we approximated by C tsz.1 8 Ω 3.2 m B 3.2 h 1.7 for 10 3, (19) in agreement with [28] for the scaing with and Ω m, but now accounting for B and h. This means that the parameter combination that is best measured by experiments probing the tsz power spectrum at mutipoes 10 3 is F (Ω m /B) 3/8 h 1/5. (20) This is the parameter combination that we wi systematicay quote. Note that simiary to [8], we opted for a repacement of the decima numbers in Eq. (19) with the cosest integers. The F -parameter differs from the parameters quoted in previous works [8, 14], i.e. Ω 3/8 m. It has the advantage of accounting for the degeneracy between the ampitude of the tsz power spectrum and the mass bias as we as the Hubbe parameter. In the most recent anaysis by Savati et a [16] and Hurier & Lacasa [9] the degeneracy with the mass bias was taken into account, however the exponent quoted in these works is in contradiction with ours. III. MAXIMUM LIKELIHOOD ANALYSIS We use the SZ information contained in the y-maps in order to set constraints on cosmoogica parameters. To do so, we sampe the parameter space via the Monte Caro Markov Chain (MCMC) method and extract the joint posterior probabiity distributions for the input parameters. The universe is assumed to be spatiay fat with an effective number of utra-reativistic species N eff = The input cosmoogica parameters that are varied are: the ampitude of the primordia power spectrum of scaar curvature perturbations A s ; the spectra index of the primordia power spectrum n s ; the optica depth at reionization τ reio ; the anguar size of the sound horizon at decouping θ s ; the reduced density parameter of baryons Ω b h 2 ; the reduced density parameter of cod dark matter Ω c h 2 and w de. Hence, h, and Ω m are obtained as derived parameters. In addition, the ampitude of the tsz power spectrum strongy depends on the mass bias B, via the pressure profie of the ICM. Therefore, B is aso treated as an input varying parameter in this anaysis. To recap, the samped parameter space is eeven dimensiona: A s, n s, τ reio, θ s, Ω b h 2, Ω c h 2, w, B, A CIB, A IR, A RS }{{}}{{} C tsz : sow param. C FG : fast param. (21)

6 ( + 1)C tsz Arnaud et a 10 PLC ( + 1)C tsz Tinker et a M 200m Tinker et a M 200m Bocquet et a M 200m Jenkins et a M 180m T08@M 1600m T08@M 200m ( + 1)C tsz Sanchez-Conde et a 14 Duffy et a 08 this work Zhao et a 09 Kypin et a ( + 1)C tsz Sanchez-Conde et a 14 Duffy et a 08 this work Zhao et a 09 Kypin et a 10 Figure 1: The tsz power spectrum computed with cass_sz in different settings: for two pressure profies (top eft), for severa hao mass function evauated at the over-density mass M 200m using Kypin et a 2010 concentration mass reation for the conversion from the viria mass to the over-density mass (top right), with the Tinker et a 2008 hao mass function evauated at the over-density masses M 1600m (bottom eft) and M 200m (bottom right) for severa concentration mass reations. The soid back ine on the bottom panes is the best-fit power spectrum to the Panck SZ data obtain from our MCMC anaysis with trispectrum (the corresponding cosmoogica parameters best-fit vaues are avaiabe on the code s webpage) and computed with Tinker et a 2008 hao mass function evauated at M 500c, so that no concentration-mass reation is needed. The and data points are shown as a andmark. with eight sow parameters which affect the ampitude of the tsz power spectrum, and three fast parameters that set the ampitude of the foregrounds. The samping is performed with Montepython [29] using the Choesky decomposition method for an optima treatment of fast and sow parameters [30]. Weak uniform priors are imposed on the varying parameters to avoid the samping of unreaistic regions of the parameter space. These priors are reported in tabe I. For each parameter, the aowed range of vaue is wide enough so that changing the upper or ower bound does not affect the posterior probabiity distribution. The data for the tota y 2 power spectrum is deduced from the y-map of the Panck survey. It is the same as the one used for the MCMC anaysis carried out by the Panck coaboration, described in [8]. We refer to this anaysis as PLC15. For competeness, the Panck 2015 measured power spectrum Ĉy2 data points and error bars min. max. min. max. min. max A s Ω bh A CIB 0 10 n s Ω ch A IR 0 10 τ reio w A RS θ s B Tabe I: Uniform priors imposed on the input parameter space for the MCMC anaysis. σ y2 are reported in tabe V. At ow mutipoes, the signa is contaminated by emissions from the gaactic dust, whie at high mutipoes, the signa is dominated by correated noise. Hence, the ikeihood cacuation is restricted to the effective mutipoe range 10 eff The effective mutipoes, eff, are at the midde of the eighteen bins spanning this interva and used in PLC15. The bin sizes were chosen by minimizing the correations be-

7 ( + 1)C tsz B = 1.1 B = 1.2 B = 1.3 B = 1.4 B = 1.5 B = ( + 1)C tsz h = 0.60 h = 0.65 h = 0.70 h = 0.75 h = ( + 1)C tsz = 0.84 = 0.82 = 0.80 = ( + 1)C tsz Ω m = 0.32 Ω m = 0.31 Ω m = 0.30 Ω m = 0.29 Ω m = ( + 1)C tsz w = 1.2 w = 1.1 w = 1.0 w = 0.9 w = ( + 1)C tsz n s = 1.00 n s = 0.95 n s = 0.90 n s = 0.85 n s = 0.80 Figure 2: Infuence of various parameters on the tsz power spectrum: the mass bias B (top eft); the reduced Hubbe parameter h (top right); the ampitude of custering (midde eft); the matter density parameter Ω m (midde right); the equation of state of dark energy w (bottom eft); and the spectra index n s (bottom right). These cacuations were carried out with cass_sz. When one parameter is varied, the other are kept constant and set to their best-fit vaues from our MCMC anaysis of the Panck 2015 SZ data (these are avaiabe on the code s webpage). The and data points are shown as a andmark. tween adjacent bins at ow mutipoe and by maximizing the signa-to-noise ratio at high mutipoe [28]. In addition to the uniform priors of tabe I, we used a rejection criterion for the ampitudes of the foreground components. Like we saw in section I, there is an upper bound for the combined foregrounds fixed by the power spectrum of the y-map obtained by projecting the confirmed custers of the Panck 2015 SZ cataogues, Ĉ RC (where stands for Resoved Custers). At each new proposed vaues for the foreground ampitudes {A CIB, A IR, A RS }, we ensure that the inequaity in (7) is satisfied for the seven mutipoe bins between eff = and eff = , otherwise the point is rejected. (Above eff = the power spectrum Ĉ RC is affected by the resoution of the y map and can not be used. Beow eff = 257.5, statistica and systematic uncertainties are

8 8 important and (7) is no onger appicabe.) The data for the power spectrum of the projected resoved custers Ĉ RC and associated error bars σ RC is aso reported in tabe V, see [8, 28] for detais. At each step of the MCMC samping, the ikeihood is computed according to n L = 1 2 χ2 with χ 2 ( ) [M C y2 1 ] a Ĉy2 a a eff (C y2 a eff aa a eff Ĉy2 a eff (22) where a, a are indices for the mutipoe bins running from a = 1 ( eff = 10) to a = 18 ( eff = 959.5), C y2 is the a eff mode y 2 power spectrum, Ĉ y2 are the data points and a eff M is the binned covariance matrix. Unike PLC15, our covariance matrix accounts for the non-gaussian contributions, i.e. auto-correations and correations between mutipoe bins, arising from the tsz trispectrum cacuation. Hence, the coefficient of the covariance matrix reative to mutipoe bins a and a reads as M aa = (σ y2 a eff ) 2 δaa + a eff ( a eff + 1) a eff( a eff + 1) 4π 2 ), T aa 4πf sky, (23) where σ y2 are the measured error bars (third coumn a eff of tabe V), f sky = 0.47 is the Panck sky coverage, and T aa is the binned trispectrum whose computation is detaied hereafter. The trispectrum T of the Compton y parameter is assumed to be dominated by the tsz effect contribution. Its computation foows the same procedure as the tsz power spectrum, with the fourth power of the y parameter in the integrand: T = dz dv d n M dn dz d n M y (M, z) 2 y (M, z) 2, (24) where the redshift and mass ranges are the same as for the power spectrum, see (8). For the eighteen mutipoe bins of our anaysis, the binned trispectrum that enters the covariance matrix (23) is T aa = a a T, (25) N a N a where a and a denotes two mutipoe bins containing respectivey N a and N a mutipoes. In figure 3 we show the trispectrum against the gaussian error. As noted by Horowitz & Sejak [14], the trispectrum argey dominates the gaussian errors unti 500. Moreover, as can be seen on the figure, the difference between the binned and unbinned trispectra is negigibe so one can safey approximate the binned trispectrum by the unbinned vaue corresponding to the effective mutipoes of the given bins. Hence, we can avoid the time consuming interpoation required for the binning. Both ampitudes of the power spectrum and trispectrum of the SZ effect are sensitive to the cosmoogica parameters h, Ω m, and to the mass bias B. At each step of the MCMC samping, when a new set of parameters is proposed, the trispectrum is computed simutaneousy to the power spectrum in order to deduce the ikeihood (22). The trispectrum contribution to the covariance matrix was considered in the anaysis by Horowitz & Sejak [14] and Savati et a [16]. However, the tsz power spectrum data points that were used in those anaysis were the one obtained in the PLC15 anaysis which did not account for the trispectrum. Reca that the tsz data points are obtained after substracting the foregrounds to the tota y 2 data. Since the foreground ampitudes are estimated from the MCMC anaysis, they are affected by the incusion of the trispectrum. In the next section we present the updated tsz data points resuting from the anaysis with trispectrum. As we sha see, the trispectrum aows for arger foreground ampitudes and hence ead to a smaer tsz ampitude essening the tension with and data at = 3000, compared to what was obtained without trispectrum in PLC15. IV. RESULTS In this section we present the resuts of three MCMC anaysis. First, we performed an anaysis without trispectrum in a fat ΛCDM universe. Second, we incude the trispectrum in the covariance matrix. Third, we consider a wcdm cosmoogy and present the constraints on the equation of state parameter of dark energy obtained from the SZ data. IV.A. Anaysis without trispectrum The Panck 2015 anaysis (PLC15) did not incude the trispectrum in the covariance matrix. They used T08 HMF evauated at the over-density mass M 500c, aong with the A10 pressure profie, aowing the foreground ampitudes as we as and Ω m to vary whie other parameters were set to their Panck 2013 CMB TT+owP best-fit vaues. The Compton parameter was integrated between z = 0 and z = 3, the mass function between M = M and M = M. Two MCMC anaysis without trispectrum were carried out by the Panck Coaboration. First, with a mass bias set to B = 1.25 (b = 0.2 in the notations of [8]). Second, with B = 1.67 (corresponding to b = 0.4). For B = 1.25, they obtained (Ω m /0.28) 3/8 = , whie for B = 1.67 they get (Ω m /0.28) 3/8 = The difference between the

9 ( + 1) ( + 1) T /4πf sky T T unbinned = σ y ( + 1)C tsz + FG tsz CIB IR RS 100 CN Ĉ y2 Ĉ y2 + σ y2 + T + σ y Figure 3: Left pane: The normaized trispectrum of the tsz effect corresponding to the best-fit mode of our MCMC anaysis of the Panck 2015 SZ data (see the code s webpage for the best-fit parameter vaues) potted against the gaussian error from the Panck 2015 anaysis (dashed ine). The binned trispectrum is shown in back and the unbinned trispectrum is shown in grey. Right pane: The best-fit modes for the tsz component (back dashed ine) and the foregrounds (grey ines) obtained from our MCMC anaysis of the Panck 2015 SZ data. The best-fit tota y 2 power spectrum appears as the soid back ine. The data points used in our anaysis are shown in grey. The gaussian contribution to the error bars corresponds to the vertica grey ines, and the error bars that account for the trispectrum contribution, which was used in our MCMC anaysis, are shown in back. centra vaues resuting from these two anaysis is due to the degeneracy between the ampitude of the tsz power spectrum and the mass bias, C tsz B 3.2 as discussed in section II. In a preiminary anaysis, we have successfuy reproduced the PLC15 constraints. For our main anaysis without trispectrum, we aso used T08 HMF evauated at the over-density mass M 500c. We have set the pressure profie to P13 and integrated between z = 0 and z = 6, and between M = h 1 M and M = h 1 M. We aowed the six cosmoogica parameters to vary aong with the mass bias B, and the foreground ampitudes. Since the mass bias is now taken as a varying parameter, the parameter (Ω m /0.28) 3/8 is not reevant anymore because the bounds on the marginaized posterior ikeihood for this parameter are determined by the bounds on the prior distribution of the bias, namey: B = 1.11 and B = Nevertheess, the marginaized posterior ikeihood for the F -parameter defined in (20), is we approximated by a gaussian and does not depend on the choices of the input prior probabiity distributions. The contours are shown in red on figure 5. The F -parameter constraint is F = ± at 68% C.L. and the foreground ampitudes are A CIB = , A IR = , A RS =. As we sha see in the next section, the width of the marginaized posterior probabiity distributions for the F -parameter and the foreground ampitudes obtained without trispectrum are argey underestimated. IV.B. Anaysis with trispectrum When the trispectrum is incuded in the covariance matrix, the width of the 68% C.L. intervas is increased drasticay, due to the enhanced degeneracy between the ampitude of the tsz power spectrum with the CIB foreground at ow mutipoes ( 400). The trispectrum aows for a smaer tsz component to the y-map, corresponding to a arger CIB and IR foreground contributions to fit the signa at 400, see figure 3. For the F -parameter as determined by our anaysis with trispectrum we obtain F = ± 0.013, (26) and the foreground ampitudes are A CIB = 0.38 ± 0.18, A IR = 2.12 ± 0.19, A RS = The best-fit modes for the tsz and combined foregrounds are shown on figure 4 against Panck y 2 data (eft pane) and / tsz measurements at = 3000 (right pane). The foreground dominate over the tsz contribution after 400, as can be seen on the eft pane. The back data points end error bars correspond to the tsz signa obtained by marginaising over the foregrounds, and propagating the corresponding errors: Ĉ tsz = Ĉ y2 C FG. On the

10 ( + 1)C tsz + FG tsz FG Ĉ y2 Ĉ RC + σ y2 + T margin. over FG ( + 1)C tsz PLC15 BF this work BF PLC15 this work Figure 4: Left pane: The best-fit modes for the tsz component (soid back ine) and the combined foregrounds (dashed grey ine) obtained from our MCMC anaysis of the Panck 2015 SZ data, extrapoated unti = The open circes correspond to projected tsz signa from the Panck 2013 custer cataogue. The continuous grey ine is the best-fit tota y 2 power spectrum. The panck 2015 data points for the y 2 power spectrum are shown in grey with the error bars that now incude the trispectrum contribution. The back data points are the updated tsz data points obtained after marginaizing the tota y 2 power spectrum over the foregrounds (determined by our MCMC anaysis). Right pane: The back data points are the updated tsz data points obtained after marginaizing the tota y 2 power spectrum over the foregrounds. The back curve is our best-fit tsz power spectrum. The grey data points and dashed ine are the resuts of the Panck 2015 SZ anaysis. We aso incuded the and tsz data points on this pot. right pane of the figure, we have aso reported the tsz data points obtained the PLC15 anaysis with the corresponding best-fit tsz mode, against our resuts. Like we said, the trispectrum increases the modeed error bars at ow mutipoes, which eads to a ower tsz component than without trispectrum and arger foreground ampitudes. The contours of the anaysis with trispectrum are shown in back on figure 5. In addition to the broadening of the marginaized posetrior probabiity distributions, we see that the IR component, which dominates around = 1000, is sensiby higher in the anaysis with trispectrum. Due to this, our best-fit tsz mode fattens significanty at arge mutipoes compared to the bestfit mode of the PLC15 anaysis. Now, the best-fit tsz mode and updated data points are in an amost perfect agreement with / data at high mutipoes. Whether this is a coincidence or an indication of the robustness of the tsz signa modeing remains an open question, as numerica simuation continue to find a tsz component that is about fifty percents arger than the tsz ampitude predicted within the hao mode [15]. Depending on the vaue of the mass bias, the tsz constraints on the F -parameter can be concied with CMB constraints.the Panck 2015 CMB TT+owP chains yieds Ω 3/8 m h 1/5 = ± In fact, if we impose this constraints on Ω 3/8 m h 1/5, on the F -parameter tsz determination, we obtain an estimation of the mass bias: B = 1.61 ± 0.16, (27) which transate into (1 b) = 0.62±0.06 in the notations of e.g. [31, 32]. This vaue is in weak tension with the estimate from most numerica simuations which tend to favor (1 b) = Nevertheess, it is consistent with the recent weak ensing constraint: (1 b) = 0.73 ± 0.10 [32]; and in a perfect agreement with the estimate extracted in the same way from custer number counts (1 b) = 0.58 ± 0.04 [2]. We aso note that for simuations where the mass dependence of the bias was investigated, it was shown that the mass bias decreases as a function of the mass towards (1 b) 0.60 for M h 1 M [33]. Our resuts seem to support such behavior of the mass bias, since the tsz signa beow 10 3 is mainy due to the most massive haos, as can be seen on figure 6. IV.C. Constraints on Dark Energy In this section, we present a new way of constraining the equation of state (EoS) for dark energy, w, based on the SZ cosmoogica data (a sky y-map) and an independent determination of the Hubbe parameter. Like we saw on figure 2, the EoS does not affect the ampitude of the tsz power spectrum in the mutipoe range of interest to us. Nevertheess, the tsz power spectrum is an indirect probe of the inear matter power spectrum and therefore it impicity probes the known degeneracy between the EoS, the Hubbe parameter and the reduced matter density. Using the oca measurement of the Hubbe parameter deduced from up-to-date time-deay cosmography measurements of quasars, h = 0.72 ± 0.03 (see

11 11 F (Ω m /B) 3/8 h 1/5 Trispectrum No Trispectrum A CIB A IR A RS Figure 5: Projected (1D and 2D) joint posterior probabiity distributions for the F -parameter and the foreground ampitudes from two MCMC anaysis, fitting the Panck y 2 power spectrum obtained from the 2015 a-sky y-map. The back contours correspond to the anaysis with trispectrum and the red contours to the anaysis without trispectrum. [35] for detais), we are abe to extract a determination of w which is competitive with CMB TT+owP+ensing constraints. We carried out a MCMC anaysis assuming a fat wcdm cosmoogy, where the six base cosmoogica parameters, the EoS for dark energy, the mass bias, and the three foreground ampitudes were samped from the uniform priors reported in tabe I. The resuting contours are shown in back on figure 7. Adding one varying parameter (the EoS) does not ater significanty the constraint on F compared to the ΛCDM case. For the wcdm anaysis we get F = ± As expected, there is no correation between the F -parameter and the EoS. Hence, measuring F does not aow for a determination of w. Nevertheess, the back contours on figure 7 ceary revea the underying correations between w,, Ω m and h. In a subsequent anaysis, we broke these degeneracies by imposing a gaussian prior on h that corresponds to the oca measurement of the Hubbe parameter [35], as we as the so-caed Normaization prior on 10 9 A s e 2τreio reported in tabe II. To further reduce the degeneracy, we aso set a gaussian prior on the optica depth at reionization τ reio = 0.06±0.01 that is a good compromise between the different measurements of the ast decade [17, 34]. Finay we fixed the vaue of the mass bias to the one favored by numerica simuations: B = The resuting contours of this anaysis with priors, dubbed tsz+h 0 on figure 7 are shown in red. The constraint on the F - parameter remains the same as for the anaysis without

12 10 12 ( + 1)C tsz [ h 1 M, h 1 M ] [ h 1 M, h 1 M ] [ h 1 M, h 1 M ] the compressed ikeihood. At each step of the MCMC, we compute χ 2 CMB = i,j= ( ) [ P P i C 1 ] ( ) ij P P j, (31) where P { Ω b h 2, A s, n s, R, A } contains the proposed vaues of the parameters and P contains the mean vaues of the parameters deduced from the wcdm TT+owP+ensing chains (reported in the first coumn of the tabe aong with the 68% standard deviations σ i which enter the covariance matrix coefficients C ij = σ i σ j D ij ). Then, the χ 2 vaue for TT+owP+ensing+H 0 Figure 6: The contributions to the tota tsz power spectrum (soid ine) coming from ow masses haos (dotted ine) and high masses haos (dashed ine) computed with cass_sz. The and data points are shown as a andmark. priors. Moreover, we get: w = , (28) for the EoS in addition to = ± 0.041, and Ω m = ± To compare these resuts with CMB constraints, we use a compressed ikeihood in order to get the TT+owP+ensing+H 0 joint posterior probabiity distribution. The technique of compressed ikeihood appied to dark energy reated data was initiated in [36]. It is based on the fact that the information contained in the Panck chains for a given mode universe (fat, curved, ΛCDM, etc.) can be efficienty recast in the form of a ow dimensiona covariance matrix for a sma set of we chosen cosmoogica parameter combinations. Then, it is not necessary to re-do the anaysis of the Panck observationa data, one can use the compressed ikeihood instead. Foowing [37], we constructed the compressed covariance matrix reative to the Panck 2015 TT+owP+ensing wcdm. The essentia parameter combinations that encapsuate the information associated with the background cosmoogy are the shift parameters, R, and the anguar scae of the sound horizon at ast scattering A. They are given by R Ω m H0 2D A (z ) /c, (29) A πd A (z ) /r = π/θ s, (30) where z is the decouping redshift, D A (z) is the comoving anguar diameter distance, r is the comoving size of the sound horizon at decouping and θ s is the corresponding comoving anguar size. In addition, we aso use parameters that carry information reative to the dynamics of perturbations, namey: A s, n s and Ω b h 2, eading to a 5x5 covariance matrix. In tabe IV, we give the normaized covariance matrix coefficients, D ij, corresponding to (aso denoted CMB+H 0 ) is given by χ 2 = χ 2 CMB + χ 2 H 0 where χ 2 H 0 corresponds to the priors on h, 10 9 A s e 2τreio and τ reio discussed previousy (see tabe II). We find w de = , = ± 0.030, and Ω m = ± 0.024, in good agreement with the tsz+h 0 anaysis and simiar width of the 68% C.L. intervas. This shows that the SZ power spectrum is a compeing probe for dark energy, amost as efficient as CMB ensing. V. SUMMARY AND CONCLUSIONS In the begining of this artice we have reviewed the cacuation of the tsz power spectrum and trispectrum based on the hao mode. We have impemented it in a new modue for CLASS and made it pubicy avaiabe on the internet. We reconcied szfast based anaysis and the Panck anaysis by identifying the source of the discrepancy: it is due to the use of the concentration mass reation for the conversion from the hao masses to the over-density mass. Then, we revisited the Panck 2015 anaysis of the tsz power spectrum and updated the tsz data with the incusion of the trispectrum in the covariance matrix, as we as varying the whoe set of cosmoogica parameters instead of ony and Ω m. We found a arger foreground contribution and a smaer tsz component to the tota y 2 power spectrum than PLC15. Our updated tsz data is now in perfect agreement with the / measurements at higher mutipoes. We saw that the parameter that is determined by the SZ data is the F -parameter, which accounts for the fact that the ampitude of the tsz power spectrum is not ony set by and Ω m but aso by the the Hubbe parameter and the mass bias. We used CMB constraints on, Ω m and h to obtain an estimate of the mass bias, which was found to be in good agreement with weak ensing estimates and in perfect agreement with the SZ number counts constraint and the prediction for arge masses haos from numerica simuations.

13 13 F (Ω m /B) 3/8 h 1/5 tsz tsz + H w Ω m h Figure 7: Projected (1D and 2D) joint posterior probabiity distribution as obtained from the tsz anaysis (a parameters varying incuding the mass bias B and the equation of state of dark energy w) and the tsz + H 0 anaysis for which we set the bias to B = 1.25 and imposed priors on h, A se 2τ reio and τ reio (see tabe II). Finay, we used the attest oca measurement of the Hubbe parameter and the Panck 2015 normaization prior in order to constrain the equation of state of dark energy from the SZ data. We shown that the tsz power spectrum is actuay a powerfu probe of dark energy, competitive with CMB weak ensing. Acknowedgment BB is very gratefu to Guiaume Hurier, Scott Kay, Fabien Lacasa, Ryu Makiya and Forian Ruppin. Appendix : SZ Data, Compressed Likeihood and Normaization Priors Gaussian priors h ± A se 2τ reio ± τ reio ± 0.01 Tabe II: Gaussian priors imposed on the parameters h, 10 9 A se 2τ reio and τ reio (in order to break the degeneracy between w de and other cosmoogica parameters in the tsz wcdm anaysis). The prior on h is from [35] (quasars time deay), the prior on A se 2τ reio was obtained from the Panck chains (see tabe III) and the prior on the optica depth is a compromise between the different measurements over the ast decade.

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