An analysis of the temperature structure of galaxy clusters by means of the thermal Sunyaev-Zel'dovich effect
aa r X i v : . [ a s t r o - ph . C O ] S e p Astronomy&Astrophysicsmanuscript no. sz-arxiv c (cid:13)
ESO 2018June 18, 2018
An analysis of the temperature structure of galaxy clusters bymeans of the thermal Sunyaev-Zel’dovich effect
Prokhorov, D. A. , , Dubois, Y. , and Nagataki, S. Korea Astronomy and Space Science Institute, Hwaam-dong, Yuseong-gu, Daejeon, 305-348, Republic of Korea Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX13RH, United KingdomAccepted . Received ; Draft printed: June 18, 2018
ABSTRACT
Aims.
Measurements of the Sunyaev-Zel’dovich (hereafter SZ) e ff ect distortion of the cosmic microwave background provide uswith an independent method to derive the gas temperature of galaxy clusters. In merging galaxy clusters the gas distribution isinhomogeneous and, therefore, the method of temperature measuring based on the SZ e ff ect should be more relevant than that basedon an X-ray emission analysis. Here we study a method for measuring the gas temperature in merging clusters by means of the SZe ff ect. Methods.
Our calculations of intensity maps of the SZ e ff ect include relativistic corrections considered within the framework of theWright formalism and utilize a cosmological numerical simulation of a merging galaxy cluster evolved with its baryon physics. Results.
We found that the gas temperature in merging clusters can be measured by means of the ratio of the SZ intensity at alow frequency (128 GHz) to that at a high frequency (369 GHz). This SZ intensity ratio permits us to reveal prominent features ofthe temperature structure caused by violent merger shock waves. Therefore, measurements of the ratio of the SZ intensities are apromising tool for measuring gas temperature in merging galaxy clusters.
Key words. galaxies: cluster: intracluster medium; relativistic processes; cosmology: cosmic microwave background
1. Introduction
Galaxy clusters are large structures in the Universe, with radiiof the order of a megaparsec. The space between galaxies in theclusters is filled with low-density (10 − –10 − cm − ) high tem-perature ( k B T ∼ −
10 keV) electron-proton plasma (for a re-view, see e.g., Sarazin 1986). Inverse Compton scattering of hotfree electrons in clusters of galaxies on the cosmic microwavebackground (CMB) radiation field causes a change in the in-tensity of the CMB radiation towards clusters of galaxies (theSunyaev-Zel’dovich e ff ect, hereinafter the SZ e ff ect; for a re-view, see Sunyaev & Zel’dovich 1980).The SZ e ff ect is important for cosmology and the study ofclusters of galaxies (for a review, see Birkinshaw 1999). It mea-sures the pressure of an electron population integrated alongthe line of sight as long as free electrons are non-relativistic.Relativistic e ff ects are significant for high temperature plasmasin galaxy clusters (see, e.g. Rephaeli 1995). A relativisticallycorrect formalism for the SZ e ff ect based on the probability dis-tribution of the photon frequency shift after scattering was givenby Wright (1979) to describe the Comptonization process of softphotons by mildly relativistic plasma.Relativistic e ff ects for the SZ e ff ect permit us to measurethe temperature of intracluster plasma (see Pointecouteau et al.1998; Hansen et al. 2002). This method is more promising formeasuring the temperature of merging and / or distant clusters ofgalaxies than those based on studies of an X-ray spectrum anal-ysis (see Pointecouteau et al. 1998). This is because the X-rayemission traces the denser component (the X-ray emission isproportional to R n ( l ) dl (where n e is the electron number den- Send o ff print requests to : D.A. Prokhorov e-mail: [email protected] sity and dl is the integrated line of sight), while the SZ e ff ect isproportional to R n e ( l ) dl ) and the cluster’s X-ray surface bright-ness strongly decreases with redshift while the SZ brightness isindependent of redshift.Using hydrodynamic simulations of galaxy clusters, Kay etal. (2008) compare the temperatures derived from the X-rayspectroscopy and the SZ e ff ect. Their method for measuring theSZ temperature is based on an equation (see Eq. 6 from Kay et al.2008) which is only valid in the Rayleigh-Jeans limit. As knownrelativistic e ff ects on the CMB intensity distortion are moresignificant at higher frequencies, Colafrancesco & Marchegiani(2010) conclude that to obtain detailed information about thecluster temperature distribution one must use high-frequencyspectral observations of the SZ e ff ect in the range 300-400 GHzand show that the SZ temperature can be extracted even for coolnon-merging galaxy clusters, such as the Perseus and Abell 2199clusters.In this paper, using the relativistically correct Wright formal-ism and results of a cosmological numerical simulation of a cooldistant merging cluster, we show how the SZ temperature canbe derived by means of multi-frequency observations of the SZe ff ect. We calculate intensity maps of the SZ e ff ect at di ff erentfrequencies and propose to use the ratio of the SZ intensities attwo frequencies to derive the SZ temperature of galaxy clusters.The layout of the paper is as follows. We describe the cosmo-logical numerical simulation of the cool distant merging clusterin Sect. 2. We calculate the SZ intensity maps at di ff erent fre-quencies in the framework of the Wright formalism in Sect. 3.We consider the ratio of the SZ intensities at two frequencies tofind a convenient method for observing the SZ temperature of D.A. Prokhorov et al.: An analysis of the temperature structure by means of the SZ e ff ect galaxy clusters in Sect. 4 and present our discussions and con-clusions in Sects. 5 and 6.
2. Numerical simulations
The galaxy cluster simulations of Dubois et al. (2010) whichwe use in this paper are run with the Adaptive Mesh Refinement(AMR) code RAMSES (Teyssier 2002). The evolution of the gasis followed using a second-order unsplit Godunov scheme for theEuler equations. The Riemann solver used to compute the flux ata cell interface is the acoustic solver using a first-order MinModtotal variation diminishing scheme to reconstruct the interpo-lated variables from their cell-centered values. Collisionless par-ticles (dark matter, stars and black hole particles) are evolvedusing a particle-mesh solver with cloud-in-cell interpolation.The simulations are performed using a re-simulation (zoom)technique: the coarse region is a 128 grid with M DM = . × M ⊙ DM resolution in a 80 h − Mpc simulation box, where h is the Hubble constant in units of 70 km s − Mpc − . This regioncontains a smaller 256 equivalent grid in a sphere of radius 20h − Mpc with M DM = . × M ⊙ DM resolution, which in turnencloses the final high resolution sphere with radius 6 h − Mpc,512 equivalent grid and M DM = . × M ⊙ DM resolution.The maximum level of refinement reached in this simulation al-lows to resolve a minimum spatial scale of 1 .
19 h − kpc.A flat Λ CDM cosmology was assumed with total matter den-sity Ω m = .
3, baryon density Ω b = . Ω Λ = .
7, fluctuation amplitude at 8 h − Mpc σ = . =
70 km s − Mpc − that corresponds tothe Wilkinson Microwave Anisotropies Probe 1 year best-fittingcosmology (Spergel et al. 2003).The resimulated region tracks the formation of a galaxy clus-ter with a 1:1 major merger occurring at z = =
3. The SZ effect from the simulated cluster
In this section, the SZ intensity maps are calculated in theframework of the relativistic Wright formalism for the simulatedgalaxy cluster.The CMB intensity change produced by the SZ e ff ect bynon-relativistic electrons considered within the framework ofthe Kompaneets approximation is (see for a review, Birkinshaw1999): ∆ I nr ( x ) = I g ( x ) y gas (1)where x = h ν/ k b T cmb , I = k b T cmb ) / ( hc ) , and the spectralfunction g ( x ) is given by g ( x ) = x exp( x )(exp( x ) − x exp( x ) + x ) − − ! . (2) Fig. 1.
The mass-weighted temperature map (in keV) of the sim-ulated cluster along the x-direction at z = nr ′ denotes the fact that Eq. (1) was obtained inthe non-relativistic limit. The Comptonization parameter y gas isgiven by y gas = σ T m e c Z n gas kT e dl (3)where the line-of-sight integral extends from the last scatteringsurface of the CMB radiation to the observer at redshift z = T e is the electron temperature, n gas is the number density of thegas, σ T is the Thomson cross-section, m e the electron mass, c the speed of light, k b the Boltzmann constant and h the Planckconstant.The CMB intensity change in the Wright formalism can bewritten in the form proposed by Prokhorov et al. (2010), and is ∆ I ( x ) = I σ T m e c Z n gas k b T e G ( x , T e ) dl , (4)where the spectral function g ( x ) is changed to the generalizedspectral function G ( x , T e ) which depends explicitly on the elec-tron temperature.The relativistic spectral function G ( x , T e ) derived in theframework of the Wright formalism is given by G ( x , T e ) = Z ∞−∞ P ( s , T e ) Θ ( T e ) x exp( − s )exp( x exp( − s )) − − x exp( x ) − ! ds (5)where Θ ( T e ) = k b T e / m e c , and P ( s , T e ) is the distribution offrequency shifts for single scattering (Wright 1979; Birkinshaw1999).There are three basic spectral features that characterize thethermal, non-relativistic SZ e ff ect signal: a minimum of its in-tensity is located at a dimensionless frequency x = .
26 ( ν = x = ν =
217 GHz), and amaximum of its intensity is located at a dimensionless frequency x = .
51 ( ν =
369 GHz).Observations of the SZ e ff ect close to the crossover fre-quency are biased due to the presence of the kinematical SZe ff ect (for a review, see Birkinshaw 1999), which is associ-ated with the peculiar velocity of the galaxy cluster. The un-known value of the peculiar velocity limits the ability to measure .A. Prokhorov et al.: An analysis of the temperature structure by means of the SZ e ff ect 3 the cluster temperature directly through the displacement of thecrossover frequency of the SZ e ff ect in the correct relativistictreatment (Colafrancesco et al. 2009).Prokhorov et al. (2010) show that the choice of frequencies x = .
26 and x = .
51 corresponding to minimum and max-imum values of the SZ intensity in the Kompaneets approxi-mation is suitable to analyze mildly relativistic electron popu-lations. Since the frequency of its maximum is located at 369GHz, observations at this high frequency should be promising toderive the temperature by means of the SZ e ff ect (Colafrancescoet al. 2010).To produce the SZ intensity maps at frequencies x = . x = .
51 we use the 3D density and temperature maps forthe simulated galaxy cluster considered in Sect. 2. We calculatedthe SZ e ff ect using the Wright formalism. The intensity maps ofthe SZ e ff ect at these frequencies derived from the simulationmaps of the gas density and temperature are plotted in Figs. 2and 3, respectively. Fig. 2.
The intensity map I / I of the SZ e ff ect at a frequency 128GHz derived from the numerical simulation in the framework ofthe Wright formalism.The morphologies of the SZ intensity simulated maps at fre-quencies x = .
26 and x = .
51 are similar. This is becausethe SZ e ff ect from cool galaxy clusters can be approximatelydescribed in the framework of the Kompaneets approximation.Note that the spectral function g ( x ) does not depend on gas tem-perature. However, we now show that the gas temperature struc-ture of a cool galaxy cluster can be derived by means of a moreaccurate analysis.
4. The ratio of the SZ intensity at frequency 128GHz to that at frequency 369 GHz
In this section, we show that the ratio of the SZ intensities atfrequencies 128 GHz to 369 GHz provides us with a convenientmethod for measuring the SZ temperature of galaxy clusters.Multi-frequency analysis of the SZ e ff ect permits us to de-rive temperature maps for galaxy clusters. Colafrancesco &Marchegiani (2010) fitted the six frequency SZ e ff ect data ( ν = ff ect and assuming, for each experimental Fig. 3.
The intensity map I / I of the SZ e ff ect at a frequency 369GHz derived from the numerical simulation in the framework ofthe Wright formalism.data point, an uncertainty of 0.1%. They present results of the fit-ting procedure used to extract the cluster temperature from a setof simulated spatially resolved spectroscopic SZ e ff ect observa-tions in di ff erent bands of the spectrum. The Perseus and Abell2199 clusters studied in Colafrancesco & Marchegiani (2010)are relaxed galaxy clusters, here we present our results of ananalysis of the temperature structure of the simulated merginggalaxy cluster.Using the relativistic Wright formalism for the SZ e ff ect wefound that the ratio of the relativistic spectral function G ( x , T e )at frequency 128 GHz to that of function G ( x , T e ) at frequency369 GHz is a monotonic function of temperature. For an isother-mal galaxy cluster the ratio of the SZ intensities is given by ∆ I ( x ) / ∆ I ( x ) = G ( x , T e ) / G ( x , T e ). Note that the ratio of theSZ intensities does not depend on temperature in the frameworkof the Kompaneets formalism.We have checked the obtained monotonical dependance us-ing the generalized Kompaneets equation derived by Challinor& Lasenby (1998) including relativistic e ff ects. Using their Eq.(28) which is valid for k B T e <
10 keV we find that the ratio ofthe SZ intensity at frequency 128 GHz to that at frequency 369GHz equals ∆ I ( x ) ∆ I ( x ) ≈ − . − . × k B T e m e c . (6)Comparing Eq. (4) from this paper and Eq. (6) from Kay etal. (2008) we find that the temperature derived from the ratio ofthe SZ intensities at frequency 128 GHz to that at frequency 369GHz within the framework of the Wright formalism is equiva-lent the Compton-averaged electron temperature for cool galaxyclusters for which Eq. (28) from Challinor & Lasenby (1998) isvalid.The ratio of the SZ intensity maps at frequencies 128 GHzand 369 GHz derived within the framework of the Wright for-malism (see Figs. 2 and 3) is shown in Fig. 4. We note thatregions of high temperature on the mass-weighted temperaturemap (see Fig. 1) corresponds to regions with low values of theSZ intensity ratio as expected from Eq. (6).Using the monotonical dependance of the SZ intensity ratioon temperature derived within the framework of the Wright for-malism we derive the SZ temperature map from the map of the D.A. Prokhorov et al.: An analysis of the temperature structure by means of the SZ e ff ect Fig. 4.
The ratio of the SZ intensity at frequency 128 GHz to thatat frequency 369 GHz for the simulated clusterratio of the SZ intensities at frequencies 128 GHz to 369 GHz.The SZ temperature map is shown in Fig. 5.
Fig. 5.
Temperature (in keV) of the simulated cluster along the xdirection at z = = ff erent redshiftsz = = = ∼ ff ect is necessary to measure the SZ tempera- ture in cool galaxy clusters with a precision of 1 keV by meansof the SZ intensity ratio.Our analysis shows that the ratio of the SZ intensities is apromising approach for measuring the temperature structure inmerging galaxy clusters.
5. Discussions
A source of bias in the observations of the SZ e ff ect could beprovided by a possibly relevant kinematic SZ e ff ect. The kine-matic SZ e ff ect arises from the bulk motion of the medium rela-tive to the CMB rest frame. In this section we propose a methodto extract the kinematical SZ e ff ect from SZ observations andalso discuss various constraints on the ability of measuring thegas temperature in distant merging galaxy clusters by means ofX-ray observations. To study the kinematical SZ e ff ect from SZ observations,Rephaeli & Lafav (1991) proposed to measure the SZ e ff ect atthe crossover frequency. The contribution of the kinematical SZe ff ect is maximal at the crossover while the thermal SZ e ff ectin the Kompaneets approximation equals zero. Using Eq. (28)from Challinor & Lasenby (1998), we calculate the ratio of theintensities of the relativistic SZ e ff ect to the kinematic SZ e ff ectat the crossover frequency. This ratio is given by I rel ( x = . I kin ( x = . ≈ − . k b T e ! / sv (7)where v is the peculiar velocity of a galaxy cluster and 300 km / sis the rms value of the peculiar velocity distribution of galaxyclusters (see Giovanelli et al. 1998).Equation 7 shows that the relativistic corrections to the ther-mal SZ e ff ect in galaxy clusters at the crossover frequency can besignificant and biases measurements of peculiar velocity by theRephaeli & Lafav method, particulary in hot clusters with tem-perature ∼
10 keV. Therefore, below we propose another methodto measure peculiar velocities. We note that the approximatefunction described the SZ relativistic corrections taken fromChallinor & Lasenby (1998) for k B T e <
10 keV has two fre-quencies at which relativistic corrections equal zero. The valuesof these frequencies are x a = .
33 and x b = .
02. Measurementsof SZ intensities at these frequencies provides us with a methodto derive the peculiar velocity. Using Eq. (28) from Challinor &Lasenby (1998), we find Z n e dl ! v c = m e c I σ T × ∆ I ( x a ) g ( x b ) − ∆ I ( x b ) g ( x a ) h ( x a ) g ( x b ) − h ( x b ) g ( x a ) (8)where the spectral function h(x) describing the kinematic SZ ef-fect is h ( x ) = x exp( x )(exp( x ) − . (9)The optical depth of the gas to Compton scattering can be ob-tained from spectral and spatial X-ray observations, and can bedetermined accurately in relaxed galaxy clusters. The proposedmethod to derive the peculiar velocity of a galaxy cluster basedon Eq. (8) is not a ff ected by biases due to relativistic correctionsof the thermal SZ e ff ect. .A. Prokhorov et al.: An analysis of the temperature structure by means of the SZ e ff ect 5 It is important that measurements of the value of (cid:16)R n e dl (cid:17) v / c permit us to extract contributions of the kinematical SZ e ff ectfrom SZ observations at frequencies of 128 GHz and 369 GHz,since the contribution of the kinematical SZ e ff ect is proportionalto (cid:16)R n e dl (cid:17) v / c . To compare methods for measuring the gas temperature in merg-ing clusters based on the SZ e ff ect with that based on X-rayemission, we produce the X-ray surface brightness map of ther-mal bremsstrahlung emission for the simulated cluster. For thesake of illustration, the normalized X-ray surface brightness mapof the simulated cluster in the [2.0-10.0 keV] band in logarith-mic scale is shown in Fig. 6, which can be obtained by modernX-ray satellites, such as XMM-Newton, Chandra, and Suzaku.The region corresponding to the highest temperature in Fig. 5 isshown by a black circle in Fig. 6. The X-ray surface brightnessof this region is two orders of magnitude smaller than the maxi-mal surface brightness value in Fig. 6, while the SZ intensity (seecontours of the SZ intensity superimposed on the X-ray surfacebrightness map, the inner contour represent 10% of the maxi-mum of the SZ intensity and the outer contour represents 1% ofthe maximum of the SZ intensity) of this region is one order ofmagnitude smaller than the maximal SZ intensity value in Figs.2 and 3. Therefore the X-ray emission traces the denser compo-nent while the SZ e ff ect provides us with a method to study morerarified gas regions. Fig. 6.
The normalized X-ray surface brightness map of the sim-ulated cluster in the [2.0-10.0 keV] band in logarithmic scale.The region corresponding to the highest temperature in Fig. 5 isshown by a black circle.The Abell 3376 cluster at redshift z = / (1 + z ) , the X-ray surface brightness for the simulated clus-ter at z = ∼ =
0. The decrease of the X-ray surface brightness withredshift constrains the ability of measuring the gas temperaturein distant galaxy clusters by means of X-ray observations.
6. Conclusions
Merging galaxy clusters are an interesting astrophysical labo-ratory for studying gasdynamic processes. Mergers of galaxyclusters are very energetic astrophysical events in which hugegravitational energy is released. In the course of a merger, a sig-nificant portion of this energy, which is carried by the gas, isdissipated by merger shock waves. This leads to a heating of thegas to higher temperatures. Studying the temperature structure ofmerging galaxy clusters is important to reveal heated gas regionsassociated with merger shock waves.Studying X-ray spectra provides us two independent ap-proaches to determine the gas temperature in galaxy clusters, thisis because the bremsstrahlung continuum spectrum depends onthe gas temperature and ratios of emission line fluxes are func-tions of the gas temperature (e.g. the ratio of the He-like to H-like iron line fluxes, see e.g. Prokhorov et al. 2009). An analy-sis of the SZ e ff ect permits us to determine the gas temperatureby studying deviations of the SZ intensity spectrum from thatderived within the framework of the Kompaneets approxima-tion. Colafrancesco & Marchegiani (2010) show that the methodbased on the SZ intensity deviations can be interesting for mea-suring the gas temperature even in cool clusters, such as thePerseus and Abell 2199 clusters, if uncertainties of observationaldata are 0.1% - 1%. In this paper, using the relativistically cor-rect Wright formalism and results of 3-D hydrodynamic numer-ical simulations of a cool distant merging cluster, we show howthe SZ temperature can be derived by means of multi-frequencyobservations of the SZ e ff ect.To produce a realistic cool distant merging cluster we use azoom cosmological simulation from Dubois et al. (2010). Wefind that the simulated cluster undergoes a violent merger atz = ff ects.We incorporate the relativistic Wright formalism for model-ing the SZ e ff ect in the numerical simulation using the algorithmproposed by Prokhorov et al. (2010). We calculate the SZ inten-sity maps at low (128 GHz) and high (369 GHz) frequencies andfind these SZ intensity maps look similar as expected for a coolgalaxy cluster. To provide a method for measuring the tempera-ture structure in cool merging galaxy clusters we propose to usethe ratio of the SZ intensities at two frequencies.We calculate the ratio of the SZ intensity at a low frequency(128 GHz) to that at a high frequency (369 GHz) for the sim-ulated merging cluster and show that this ratio is a promisingmethod to measure the SZ temperature in galaxy clusters. Thecalculated map of the SZ temperature shows that main features(such as “arc-like” structures) of the temperature structures ofmerging clusters may be revealed by using this method.The next generation SZ e ff ect experiments discussed inColafrancesco & Marchegiani (2010) are needed to reach therequired high sensitivity with the purpose of being independentin measuring gas temperature in galaxy clusters. Experimental D.A. Prokhorov et al.: An analysis of the temperature structure by means of the SZ e ff ect configurations which are basically the same as those of theMillimetron project (see http: // / millimetron / )will have the power to measure the gas temperature via the SZe ff ect in any cluster, including cool ones. Acknowledgements.
We are grateful to Florence Durret for valuable discussionsof X-ray emission of the Abell 3376 cluster and to Sergio Colafrancesco forvaluable discussions of experimental configurations for Millimetron.
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