The contribution of the kinematic Sunyaev-Zel'dovich Effect from the Warm Hot Intergalactic Medium to the Five-Year WMAP Data
Ricardo Genova-Santos, Fernando Atrio-Barandela, Jan Muecket, Jochen Klar
aa r X i v : . [ a s t r o - ph . C O ] M a y The contribution of the kinematic Sunyaev–Zel’dovich Effect from the WarmHot Intergalactic Medium to the Five-Year
Wilkinson Microwave Anisotropy Probe
Data
R. G´enova-Santos , , F. Atrio-Barandela , J.P. M¨ucket & J.S. Klar ABSTRACT
We study the contribution of the kinematic Sunyaev–Zel’dovich (kSZ) effect, gen-erated by the warm-hot intergalactic medium (WHIM), to the cosmic microwavebackground (CMB) temperature anisotropies in the Five-Year
Wilkinson MicrowaveAnisotropy Probe (WMAP) data. We explore the concordance ΛCDM cosmologicalmodel, with and without this kSZ contribution, using a Markov chain Monte Carloalgorithm. Our model requires a single extra parameter to describe this new compo-nent. Our results show that the inclusion of the kSZ signal improves the fit to the datawithout significantly altering the best-fit cosmological parameters except Ω b h . Theimprovement is localized at the ℓ &
500 multipoles. For the best-fit model, this extracomponent peaks at ℓ ∼
450 with an amplitude of 129 µ K , and represents 3.1% ofthe total power measured by the Wilkinson Microwave Anisotropy Probe . Neverthe-less, at the 2 σ level a null kSZ contribution is still compatible with the data. Partof the detected signal could arise from unmasked point sources and/or Poissonianlydistributed foreground residuals. A statistically more significant detection requires thewider frequency coverage and angular resolution of the forthcoming Planck mission.
Subject headings: cosmic microwave background. Cosmology: theory. Cosmology:observations
1. Introduction
Baryons represent a small fraction of the total mass–energy budget of the Universe and donot play a predominant role in its evolution. They are the only matter component that has beenidentified directly. The baryon fraction of the Universe has been determined at different redshiftsthrough a variety of methods: Ω b h = 0 . ± .
002 (Burles, Nollett, & Turner 2001) from Big Instituto de Astrof´ısica de Canarias, V´ıa L´actea, s/n. 38200 La Laguna, Tenerife, Spain; email:[email protected] Astrophysics Group, Cavendish Laboratory, University of Cambridge CB3 OHE, UK F´ısica Te´orica, Universidad de Salamanca, 37008 Salamanca, Spain; email: [email protected] Astrophysikalisches Institut Potsdam. D-14482 Potsdam, Germany; email: [email protected]; email: [email protected] b h > .
021 (Rauch et al. 1997) from the Ly α forest and Ω b h =0 . ± . z = 0 is Ω b h =0 . ± .
003 (Fukugita et al. 1998), indicating that half of the baryons in the local Universe arestill undetected.Cosmological simulations of large-scale structure formation (Petitjean et al. 1995; Zhang et al.1995; Hernquist et al. 1996; Katz et al. 1996; Theuns et al. 1998; Dav´e et al. 1999) have shown thatthe intergalactic gas has evolved from the initial density perturbations into a complex network ofmildly nonlinear filaments in the redshift interval 0 < z <
6. With cosmic evolution, a significantfraction of the gas collapses into bound objects; baryons in the intergalactic medium (IGM) arein filaments containing Ly α systems with low HI column densities and, at low redshifts, shock-confined gas with temperatures T e ∼ . δ b ∼ −
50 (Dav´e et al. 2001;Cen & Ostriker 2006). An important fraction of the missing baryons could be located in this webof shock-heated filaments, called “warm/hot intergalactic medium” (WHIM). Observational effortsto detect this “missing baryon” component range from looking at its emission in the soft X-raybands (Zappacosta et al. 2005), ultraviolet absorption lines in the spectra of more distant sources(Nicastro et al. 2005) or Sunyaev–Zel’dovich (SZ) contributions in the direction of superclustersof galaxies (G´enova-Santos et al. 2008; see Prochaska & Tumlinson 2008 for a review). Indirectsearches of the WHIM using the SZ effect have been inconclusive. The SZ imprint due to galaxyclusters in
WMAP data is well measured (Atrio-Barandela et al. 2008b), but when this componentis removed no signal associated to the WHIM remains (Hern´andez-Monteagudo et al. 2004).Atrio-Barandela & M¨ucket (2006) developed a formalism to account for the contribution of theIGM/WHIM to the CMB anisotropies via the thermal SZ (tSZ, Sunyaev & Zeldovich 1972) effect.The main assumption was that missing baryons were distributed as a diffuse gas phase outsidebound objects. Its filamentary structure was assumed to be described by a log–normal distributionfunction, which accurately models a mildly non-linear density field when the velocity field remainsin the linear regime (Coles & Jones 1991). The predicted power spectrum of the tSZ effect peaksat ℓ ∼ ℓ ∼ Wilkinson MicrowaveAnisotropy Probe ( WMAP ) data. We use a Markov chain Monte Carlo (MCMC) method to samplethe parameter space of the concordance ΛCDM model with and without a kSZ component todetermine if there is a statistically significant contribution. Briefly, in Sections 2 and 3 we describethe model and the numerical implementation of the MCMC, and in Section 4 we present our resultsand summarize our main conclusions. 3 –
2. The thermal and kinematic SZ effect from the IGM/WHIM.
The tSZ effect is the weighted average of the electron pressure along the line of sight; the kSZis proportional to the column density of the free electrons along the line of sight ˆ n , weighted by theradial component of their peculiar velocities: (cid:18) ∆ TT (cid:19) tSZ (ˆ n ) = G ( ν ) k B σ T m e c Z dl n e T e , (cid:18) ∆ TT (cid:19) kSZ (ˆ n ) = σ T c Z dl n e ( ~v e · ˆ n ) . (1)In these expressions, T e , n e , v e are the electron temperature, density and peculiar velocity, respec-tively, k B is Boltzmann constant, σ T Thompson cross section, m e c the electron annihilation energy, c is the speed of light and G ( ν ) is the frequency dependence of the tSZ effect. In the Rayleigh–Jeansregime, G ( ν ) ≈ − WMAP frequencies. For an isothermal clusterthe tSZ to kSZ ratio is: (∆ T tSZ / ∆ T kSZ ) ≃ G ( ν )( T e /
10 keV)(300 km s − /v e ). The temperatureof the IGM is much lower than in galaxy clusters even in the shock-heated WHIM and the kSZcontribution could become comparable to that of the tSZ effect.In Atrio-Barandela & M¨ucket (2006) and Atrio-Barandela et al. (2008a) we computed the tSZand kSZ temperature anisotropies generated by the IGM assuming that baryons are distributedas in a log–normal random field. The log–normal distribution was introduced by Coles & Jones(1991) as a model for the non-linear distribution of matter in the Universe. The number densityof electrons n e can be obtained from the baryon distribution assuming ionization equilibrium be-tween recombination and photoionization and collisional ionization. In the conditions valid for thephotoionized IGM the gas is almost completely ionized. The correlation function of the tSZ tem-perature anisotropy generated by two filaments located at two different redshifts along two lines ofsight with an angular separation α is dominated by the spatial variations of the electron pressureat nearby locations (Atrio-Barandela & M¨ucket 2006). Within the small angle approximation: C ( α ) = (cid:20) k B σ T m e c G ( ν ) (cid:21) Z z f dz (cid:18) dldz (cid:19) n e ( z ) T ( z ) e γ ( γ − ( z ) [ e γ Q ( α,z ) − . (2)In this expression, ∆ ( z ) is the variance of the baryon density field, which is related to the darkmatter power spectrum P ( k ) by:∆ ( z ) = D ( z ) Z d k (2 π ) P ( k )[1 + x b ( z ) k ] (3)where D ( z ) is the growth factor of matter density perturbations and x b is the comoving Jeanslength. Also, Q ( α, z ) = D ( z )2 π Z ∞ P ( k ) k dk [1 + x b ( z ) k ] j ( kα ) , (4)where j is the zeroth order spherical Bessel function. The integration in eq. (2) extends to thehighest redshift z f . 4 –Similar arguments can be used to compute the correlation function of the kSZ effect due totwo filaments located at distances l and l ′ along two lines of sight separated by an angle α (seeAtrio-Barandela et al. (2008a) for further details): C ( α ) = f σ c Z l ( z f ) o Z l ′ ( z f ) o dldl ′ V B ( R ( z )) D v ( z ) V B ( R ( z ′ )) D v ( z ′ ) h n e ( l ˆ n, z ) n e ( l ′ ˆ n ′ , z ′ ) i , (5)where V B ( R ( z )) denotes the mean bulk velocity of a sphere with electron density n e and radius R ( z ), the comoving distance from the observer to a filament at redshift z , D v ( z ) is the velocitylinear growth factor and f b is the fraction of the baryons in WHIM filaments.The correlation functions given in eqs (2) and (5) differ in two significant aspects: (a) Whilethe tSZ effect is cumulative and several filaments along the line of sight add linearly to the totaleffect, eq. (5) requires all filaments to be moving with the same velocity. This is the physical reasonwhy only the bulk flow velocity contributes to the effect. If there are several filaments, in a givendirection, with different velocities the net effect will not be ∝ ( n e V B ) but ∝ ( n e V B ) / . (b) The IGMis usually described by a polytropic equation of state T ∝ n γ − , i.e., high dense regions contributemore to the tSZ effect since their temperature is higher. In eq. (2) we then integrate all scales outto the largest overdensity δ max that is well described by the log–normal model. Otherwise we wouldbe including small contributions from gas in bound objects and not described by the log–normalmodel. This restriction is not necessary for the kSZ effect. In the log–normal approximation,velocities are in the linear regime and they are not correlated with matter overdensities. Then,instead of introducing an arbitrary cut-off at overdensity δ max , we extend our integration to alloverdensities, i.e., to all baryons, and introduce a factor f b to take into account the actual fractionof baryons in the WHIM; we could then, in principle, constrain this fraction directly from the data.Eqs (2) and (5) can be inverted to give respectively the radiation power spectra C tSZ ℓ and C kSZ ℓ at each multipole ℓ . These calculations are computationally expensive since, to get accurateresults, we use line-of-sight separations of 30 ′′ and redshift intervals of ∆ z = 0 .
005 up to z = 0 . z = 0 . z f = 1, where the contribution becomes negligible. The final tSZ powerspectrum depends on cosmological and physical parameters: the cut-off scale δ max , the amplitudeof the matter density fluctuations on spheres of 8 h − Mpc, σ , the mean gas temperature, T , thegas polytropic index, γ , and the mean gas temperature at reionization, T m . The kSZ contributiondepends on the Jeans length x b , σ and f b . Since the product γT m fixes x b then the kSZ contributionrequires less parameters than the tSZ.In Figure 1a we compare CMB power spectrum of the concordance ΛCDM model with thekSZ and tSZ WHIM contributions. The parameters of the concordance model are those of thebest-fit 5 year WMAP data. The SZ contributions are calculated using: σ = 0 . γ = 1 . γT m ≈ . × K and f b = 0 .
5. First, the relative amplitude of the thermal and kinematiccontributions depends on model parameters; the kSZ effect could be larger than the tSZ effectsince the average temperature of the IGM is rather low. Second, since the IGM is not isothermal,the tSZ contribution is dominated by the mildly overdense regions, which subtend smaller angles 5 –than the filaments themselves, so its power is shifted to higher ℓ . In Figures 1b,c we show thecontribution of different redshift intervals to the total power for each of the WHIM contributions.In both cases, the main contribution comes from z = 0 to 0 .
4. Contributions from higher redshiftsdecrease rapidly.The purpose of this article is to search for a IGM/WHIM contribution in the 5 year
WMAP data, which are sensitive to ℓ . ℓ ∼ < ℓ < ℓ ∼ ≤ µ K at 95% confidence level. The expected tSZ contribution at WMAP scales is then negligible. For this reason, we shall consider only a kSZ component. This restrictionsimplifies our study since the kSZ power spectrum depends only on three parameters: σ , γT m and f b . In Figure 2 we plot kSZ power spectra for different values of σ and γT m with f b = 0 .
5. Aregression fit of the variation of the maximum amplitude of the kSZ power spectrum with each ofthese three parameters permits us to write the following scaling relation: A kSZ ∼ ( γT m ) − σ f . (6)While the amplitude depends on the three parameters, the location of the maximum is onlyweakly dependent on γT m . Since f b is a multiplicative factor that affects only the amplitude butnot the shape or position of the maximum, variations on f b and σ cannot be distinguished. ThekSZ effect is then effectively described by two parameters: one cosmological ( σ ) and one physical( γT m ) determining the Jeans length.Figure 1c also shows that most of the contribution comes from very low redshifts. Similarscaling relations hold for the tSZ contribution of clusters of galaxies, indicating that we couldgenerate arbitrary large contributions. In semi-analytical estimates, the abundance of clusters isgiven by the Press–Schechter formalism (Atrio-Barandela & M¨ucket 1999; Molnar & Birkinshaw2000) and a large tSZ effect is obtained by arbitrarily increasing the number of clusters. Thescaling relation given in eq. (6) is limited by the validity of eq. (5); if filaments overlap along theline of sight, it overpredicts the signal. Since the kSZ effect is just an integral along the line ofsight of all electrons with a coherent peculiar motion, an order of magnitude estimate of the effectis (Hogan 1992): (∆ T ) kSZ ≈ µ K (cid:18) δ b (cid:19) (cid:18) f b . (cid:19) (cid:18) L
30 Mpc (cid:19) (cid:18) V B
600 km / s (cid:19) (7)where δ b is the overdensity of a typical filament and L is the coherence scale of a motion withamplitude V B . As explained in Atrio-Barandela et al. (2008a), the distribution of the effect is ratherskewed and 9% of all lines of sight will produce an effect 1.8–6 times larger than the estimate fromeq. (7). In the concordance model, bulk flow velocities of ∼
200 km/s are typical for volumesof R ∼
100 Mpc/h radius. Kashlinsky et al. (2008, 2009) reported a bulk flow of amplitude600–1000 km/s on a scale of 300 Mpc/h that could give a much larger contribution. A significantfraction of the temperature decrement of − µ K detected in the intercluster medium of the Corona 6 –Borealis Supercluster by G´enova-Santos et al. (2005) could be due to thermal and kinematic SZcontributions. However, that decrement is in a direction almost perpendicular to the bulk flowcited above and, because of its orientation with respect to the line of sight, this flow would notinduce a significant kSZ contribution in Corona Borealis. Only a smaller kSZ contribution couldexist due to peculiar motions on the scale of the supercluster itself.
3. Markov chain Monte Carlo parameter estimation
To explore the parameter space of ΛCDM models with and without a kSZ contribution, we usedthe April 2008 version of the cosmomc package (Lewis & Bridle 2002). This software implementsan MCMC method that performs parameter estimation using a Bayesian approach. When the kSZcontribution is included, precomputed C kSZ ℓ are added at each step of the chain to the theoreticalpower spectrum, computed with camb (Lewis et al. 2000) for each set of ΛCDM cosmologicalparameters. The model is compared with the data using the likelihood code supplied by the WMAP team (Dunkley et al. 2009).We considered the concordance ΛCDM model, defined by a spatially flat Universe with colddark matter (CDM), baryons and a cosmological constant Λ. The relative contributions of thesecomponents are given in units of the critical density: Ω cdm , Ω b and Ω Λ = 1 − Ω m , where Ω m =Ω cdm + Ω b is the total matter density. The specific parameters used in the analysis were thephysical densities Ω b h and Ω cdm h where h = H /
100 km s − Mpc − is the normalized Hubbleconstant. In order to minimize degeneracies, instead of h we used the angular size of the firstacoustic peak θ , i.e. the ratio of the sound horizon to the angular diameter distance to lastscattering (Kosowsky et al. 2002). We considered adiabatic initial conditions and assumed aninstantaneous reionization parameterized by its optical depth to Thomson scattering up to themoment of decoupling τ . The initial fluctuation spectrum was parameterized as a power law, P ( k ) = A S (cid:18) kk c (cid:19) n s − , (8)where A S is the amplitude at k c = 0 .
05 Mpc − and n s the spectral index. Since the effect of f b onthe kSZ power spectrum is indistinguishable from that of σ we fix that parameter to a given value.Thus, the kSZ signal is defined by σ , which is derived from the previous cosmological parameters,and γT m . In summary, our cosmological model with the kSZ component has seven degrees of free-dom and is described by the parameterization Θ = [Ω b h , Ω cdm h , θ, ln(10 A S ) , n s , τ, log( γT m )].In Table 1 we give the flat priors imposed on each parameter, the initial values and distributionwidths. In addition, we used a top-hat prior 10 < t <
20 Gy for the age of the Universe. Whenrunning the chains we included only the 5 year
WMAP data (Hinshaw et al. 2009) and no otherCMB or cosmological datasets.At each step in the chain, a set of initial values for the cosmological parameters and for γT m are generated. The ΛCDM radiation power spectrum and σ are computed for the basic model. 7 –When introducing the kSZ component into the model, we would have to compute the values of C kSZ ℓ at each step of the chain. However, as the computation is very demanding, we only calculate kSZpower spectra on a 2D σ − γT m grid. We built a grid with a bin width ∆ σ = 0 .
05 in the interval σ = [0 . , .
90] and ∆( γT m ) = 1 × K in the interval γT m = [1 , × K (plus interleavedvalues at 0.8, 0.9 and 1.3 × K). At some random points in the parameter space we checked thatthe interpolated and exact spectrum did not differ by more than 7% in the interval ℓ ∼ σ = [0 . , . γT m = [1 . , . × K, the bin widths were ∆ σ = 0 .
01 and ∆( γT m ) = 0 . × K,respectively. With this resampling, the differences between interpolated and exact power spectrawere lower than ∼
2% in the same ℓ range. The kSZ power spectrum is obtained at each step ofthe chain by logarithmic interpolation of the precomputed power spectra in the four closest nodesof this grid.Initially, between two consecutive steps in the chain, we added to each parameter a randomincrement drawn from a Gaussian distribution with a standard deviation equal to the value listedin Table 1 multiplied by 2.4. Since we found that γT m was strongly correlated with Ω b h andln(10 A S ), we reran the chains using the eigenvalues of the covariance matrix of the parameters(computed with the getdist facility which is part of the cosmomc package) as distribution widths.We ran nine independent chains, with a total number of 150 000 independent samples, when no kSZis included. With kSZ we fixed the baryon fraction in WHIM at five different values f b =0.3, 0.4,0.5, 0.6 and 0.7 (we also tried to constrain a model with f b as a free parameter, but the degeneraciesdid not allow us to obtain reliable results). In each case we ran eight independent chains of similarsizes with a total number of samples &
225 000. We used the R statistic (Gelman & Rubin 1992)as a convergence criterion. All our parameters have R well below 1.2: R ≈ .
008 for log( γT m ) and ≈
4. Results and discussion
In Figures 3a,b,c we plot the mean 1D likelihoods for γT m , σ and the more informative rmstemperature fluctuation introduced by the kSZ signal defined as: < ∆T kSZ2 > / = " π X ℓ =2 (2 ℓ + 1) C kSZ ℓ / . (9)The different lines correspond to different baryon fractions. Since the kSZ parameters affect theamplitude of the spectrum most significantly (see Figure 2) then, for each f b the maximum 1Dlikelihood of γT m shifts but the kSZ signal remains roughly constant (see Figure 3c). This highdegeneracy indicates that 5 year WMAP data is insensitive to the fraction of baryons in the WHIMand hereafter we will quote results for f b = 0 . σ and 2 σ contours, whereas the white dots indicate the positions of the maximum likelihood in 8 –the full parameter space. Not unexpectedly, their positions are slightly shifted with respect to the2D likelihood maxima. In fact, Markov chains estimate confidence intervals more precisely thanlocate the maximum of the likelihood, the bias being larger the greater the model dimensionality(Liddle 2004). Figure 3d shows a noticeable degeneracy between σ and log( γT m ). Within the1 σ confidence region A kSZ varies between ∼ µ K and ∼ µ K . Regions with high values of σ and low values of log( γT m ) are strongly ruled out since they overpredict the kSZ contribution.At the 2 σ level absence of a kSZ contribution is compatible with the data. This is also seen inFigures 3d,e. Models with kSZ prefer a lower value of σ (see Figure 3b), since adding kSZ requiresless primordial CMB power. Figure 3e(f) shows that Ω b h is (inversely) proportional to h T i / (log[ γT m ]). Physically, kSZ adds power mainly at ℓ ∼ b h increases.The parameters that maximize the likelihood are given in Table 2. The 1 σ confidence intervalshave been computed from the 1D mean likelihood distributions. The values for the concordancemodel agree well with those given by the WMAP team (Dunkley et al. 2009), indicating that weare correctly exploring the parameter space. The model with kSZ has larger error bars, as weare fitting the same data with an extra degree of freedom. The best-fit model has a kSZ powerspectrum with a maximum amplitude of A kSZ = 129 +138 − µ K centred at ℓ ≈ γT m = 1 . +0 . − . × K. The rms temperature fluctuation is h T i / = 19 +7 − µ K, very significantcompared with the contribution from the primordial CMB, h T i / = 110 µ K. None of theparameters except Ω b h differ by more than 1 σ from those of the concordance model. The modelwith kSZ gives a higher value for Ω b h . This reinforces the slight discrepancy between the valuesobtained from CMB temperature anisotropies and from BBN. As indicated above, adding kSZboosts the value of Ω b h since it reduces the height of the second acoustic peak. The difference in σ is also notable even if the confidence regions overlap at the 1 σ level. Our estimate is closer to thevalues obtained from the number density of galaxy clusters and the optical or X-ray cluster massfunctions. For example, Voevodkin & Vikhlinin (2004) found σ = 0 . ± .
04 from the baryonmass fraction of a sample of 63 X-ray clusters. After compiling cluster determinations since 2001Hetterscheidt et al. (2007) obtained σ = 0 . ± . σ = 0 . ± . WMAP data. We plot the best-fit power spectra for the models with and without kSZ after subtracting the
WMAP band powers. The kSZ model achieves a better fit to the experimental data points in themultipole range ℓ ∼ χ per ℓ -band for the model without and with kSZ (note that these valueswere calculated, just for illustrative purposes, by applying the χ statistics to the binned best-fittheoretical power spectra, and not by the WMAP likelihood code). This ratio is overall > ℓ ∼ χ reduces by ∆ χ = − .
3, from 2661.05 to 2657.83. 9 –Evaluating the statistical significance of this result requires taking into account the number ofdegrees of freedom, given by the difference between the number of independent data points N andof model parameters k . The WMAP likelihood is computed as a sum of different temperatureand polarization terms (Dunkley et al. 2009). In this computation 968 points correspond to theTT power spectrum at ℓ = 33–1000 and 427 to the TE cross-correlation at ℓ = 24–450. For low( ℓ ≤
23) multipoles the likelihood associated with the TT, TE, EE and EB correlations is evaluateddirectly from 1170 pixels of the temperature and polarization maps. In total N = 2565. Addingthe kSZ contribution raises the number of parameters from k = 6 to k = 7. Note that the χ perdegree of freedom also decreases when the kSZ component is included, from χ = 1 . k + χ or the more conservative Bayesianinformation criterion (Schwarz 1978) BIC = χ + k ln N . Since we are adding a single parameter, weobtain ∆AIC = − .
2, which is marginal evidence in favor of introducing this new parameter while∆BIC = 4 . σ level. Even if at present model selection does not clearly favors the kSZ contribution,it is a model well motivated physically and, as remarked by Linder & Miquel (2008), model selectionneeds to involve physical insight. A statistically more significant measurement would require a widerfrequency coverage and angular resolution, as will be provided by the forthcoming Planck mission.The kSZ contribution represents 3.1% of the power measured in 5 year
WMAP data. This con-tribution is larger than expected; typical filaments would give rise to ∆ T kSZ ∼ µ K (eq. 7). Wewould need a more accurate (numerical) model to establish which would be the physical conditionsthat give rise to the quoted kSZ signal. Numerical simulations over large cosmological volumes arecomputationally very expensive because they require high resolution over most of the volume. Infact, spatial resolution is usually an important limitation. Adaptive mesh refinement (AMR) tech-niques are efficient at describing the dynamics of gas in the high dense regions but their resolutionfalls sharply in less dense environments (Refregier & Teyssier 2002), while smooth particle hydro-dynamics (SPH) codes do not yet resolve scales as small as the Jeans length (Bond et al. 2005), thateffectively dominate the WHIM SZ contribution (Atrio-Barandela & M¨ucket 2006). Hallman et al.(2007) used the AMR Enzo code (O’Shea et al. 2005) to simulate a (512 Mpc/h) volume includingunbound bas, and found that one-third of the SZ flux in a 100 square-degree region comes fromobjects with masses below 5 . × M ⊙ and filamentary structures made up of WHIM gas, aconclusion similar to that reached by Hern´andez-Monteagudo et al. (2006). Hallman et al. (2009)focused in the low-density WHIM gas by restricting their analysis of the same AMR simulation toregions with temperatures in the range 10 − K and overdensities δ <
50, even though theirlow mass halos were not yet resolved gravitationally. They computed the radiation power spectrumfrom that simulation and found a similar shape to that of Figure 1, although with a much smalleramplitude. The high amplitude we found could be an indirect confirmation of large scale peculiarmotions reported in Kashlinsky et al. (2008) and Kashlinsky et al. (2009). If bulk flows of this am- 10 –plitude are very common, filaments at higher redshift could also contribute significantly, therebyincreasing the total signal. Also, we must take into account a possible foreground contamination:power spectra with a single maximum are also good models for Poissonianly distributed foregroundresiduals or unresolved point sources. Since the
WMAP window function exponentially dampspower at ℓ & C ℓ = const. spectrum with this window function peaks at ℓ <
600 and has a shape similar to that of the spectrum described above. Foreground substractioncan differ in the range 3–5 µ K depending on the method (Ghosh et al. 2009) and some residualscould be present on the data at that level. To distinguish the WHIM kSZ and foreground signalswe will have to include the frequency dependence, amplitude and location of the maximum for eachcomponent. In the simplest model with kSZ and one foreground, we would need four parametersto model both components so an analysis of all components is unfeasible with
WMAP
WMAP data. Thislarge amplitude is difficult to account for in the concordance ΛCDM model, where filaments areexpected to have . µ K contributions. It could be an indication that large scale flows are rathercommon. We cannot rule out that part of this contribution could be due to unmasked point sourcesand/or foreground residuals. The
Planck satellite, with its wider frequency coverage, lower noiseand different scanning strategy, is well suited for detecting the IGM/WHIM thermal and kinematiccontributions with much higher statistical significance and to distinguishing this signal from otherforeground contributions.We are thankful to Rafael Rebolo for enlightening discussion and suggestions. This work issupported by the Ministerio de Educaci´on y Ciencia and the “Junta de Castilla y Le´on” in Spain(FIS2006-05319, programa de financiaci´on de la actividad investigadora del grupo de ExcelenciaGR-234).
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This preprint was prepared with the AAS L A TEX macros v5.2.
13 –Fig. 1.— (a) Intrinsic CMB radiation power spectrum (solid line) and the IGM/WHIM tSZ andkSZ (dashed lines) contributions. (b) Contributions of redshift intervals to the total tSZ spectrum.From left to right, solid lines correspond to contributions from the following redshift intervals:[0,0.005], [0.005,0.05], [0.05,0.1], [0.1,0.2], [0.2,0.3] and [0.3,0.4]. (c) Redshift contribution to thekSZ effect. Solid lines give the contribution for z = 0 to 0 . z = 0 .
01; dotted linesgive the contributions for z = 0 . . z = 0 .
1. Dashed lines show the total power spectrafrom all the redshift intervals. The SZ contributions are calculated in all cases using σ = 0 . γ = 1 . γT m = 1 . × K and f b = 0 . σ = 0 .
80 anddifferent values of γT m (solid lines), and for a fixed γT m = 1 . × K and different values of σ (dotted lines). The fraction of baryons stored in the form of WHIM has been fixed at f b = 0 . γT m , σ and the rms temperatureanisotropy contribution of the kSZ component. Each line correspond to a different fraction ofbaryons in the form of WHIM. In (b) we also plot the likelihood for the concordance (no kSZ)model. Bottom: mean 2D likelihoods for three different combinations of parameters. Solid linesdepict the 1 σ and 2 σ confidence regions, and the thick dots indicate the positions of the likelihoodmaxima found in the full parameter space. Dotted curves in (d) represent levels with the same kSZamplitude. 15 –Basic Parameter Starting Distributionparameter limits points widthΩ b h (0.005, 0.1) 0.0223 0.001Ω cdm h (0.01, 0.99) 0.105 0.01100 θ (0.5, 10) 1.04 0.002ln(10 A S ) (2.7, 4.0) 3.0 0.01 n s (0.5, 1.5) 0.95 0.01 τ (0.01, 0.8) 0.09 0.03log( γT m ) (3.778, 4.740) 4.096 0.006Table 1: For each independent parameter, we indicate its range of variation (top-hat prior), theinitital values and the estimated distribution widths initially used by the chain to vary each pa-rameter.Fig. 4.— Top: best-fit power spectra for the models with (solid line) and without (dotted line)kSZ, after subtracting the WMAP experimental band powers. The vertical lines represent the 1 σ error bars of the 5 year WMAP data. Bottom: ratio of the χ in ℓ -bands of width ∆ ℓ = 15 for amodel without and with a kSZ component. 16 –Parameter CMB alone CMB and kSZΩ b h . +0 . − . . +0 . − . Ω cdm h . +0 . − . . +0 . − . θ . +0 . − . . +0 . − . ln(10 A S ) 3 . ± .
05 3 . +0 . − . n s . ± .
02 0 . ± . τ . +0 . − . . +0 . − . γT m (10 K) 1 . +0 . − . Ω Λ . +0 . − . . +0 . − . Ω m . +0 . − . . +0 . − . σ . +0 . − . . +0 . − . z re . +1 . − . . +1 . − . H . +2 . − . . +6 . − . h T i / ( µ K) 19 . +6 . − . A kSZ ( µ K ) 128 . +138 . − . χ f b = 0 .
5. The upper and lower (belowthe line) sets correspond to independent and derived parameters, respectively. The central valueshave been derived from the sample with the minimum χ in the chains (also shown in the table),whereas the 1 σσ