Issue 
A&A
Volume 520, SeptemberOctober 2010



Article Number  A31  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/200913115  
Published online  24 September 2010 
On the ability of the spectroscopic SunyaevZeldovich effect measurements to determine the temperature structure of galaxy clusters
S. Colafrancesco^{1,2}  P. Marchegiani^{2,3}
1  ASI, V.le Liegi 26, 00100 Roma, Italy
2 
INAFOsservatorio Astronomico di Roma
via Frascati 33, 00040 Monteporzio, Italy
3 
Dipartimento di Fisica, Università di Roma La Sapienza, P.le A. Moro 2, Roma, Italy
Received 13 August 2009 / Accepted 14 February 2010
Abstract
Aims. We explore the ability of spatially resolved
spectroscopic measurements of the SZ effect (SZE) to determine the
temperature profile of galaxy clusters. We derive a general formalism
for the thermal SZE in galaxy clusters with a nonuniform
temperature profile that can be applied to both coolcore clusters and
noncoolcore clusters with an isothermal or nonisothermal temperature
structure.
Methods. We develop an inversion technique by means of which the
electron distribution function can be extracted from spectroscopic
SZE observations over a wide frequency range. We study the fitting
procedure to extract the cluster temperature from a set of simulated
spatially resolved spectroscopic SZE observations in different
bands of the spectrum from 100 to 450 GHz.
Results. We present our analysis results for three different
cluster prototypes: A2199 with a lowtemperature cool core, Perseus
with a relatively hightemperature cool core, and Ophiuchus with an
isothermal temperature distribution. These results indicate both the
precision of the SZE observations and the optimal frequency bands
required to determine the cluster temperature with similar or better
accuracy than that obtainable from Xray observations. The precision of
SZEderived temperature is also discussed for the outer regions of
clusters. Using our method, we also study the possibility of extracting
the parameters characterizing the nonthermal SZE spectrum of the
relativistic plasma contained in the lobes of radio galaxies as well as
the spectrum of relativistic electrons cospatially distributed with the
thermal plasma in clusters that exhibit nonthermal phenomena.
Conclusions. We find that the next generation
SZE experiments, which will have both spectroscopic capabilities
with moderate resolution of a few to tens GHz and imaging
capabilities with spatial resolution of tens of arcsec up to arcmin,
can provide precise temperature distribution measurements over a wide
range of radial distances for galaxy clusters even out to high
redshift.
Key words: cosmology: theory  galaxies: clusters: general
1 Introduction
Galaxy clusters are the largest gravitationally bound container of diffuse baryons in the universe. These baryons accumulate in the cluster gravitational potential, which is dominated by cold dark matter and show, in their hotter phase (i.e. the hot intracluster medium  hereafter ICM  of temperature K), a complex temperature distribution in the cluster atmospheres as indicated by observations (see, e.g., Arnaud 2005, for a review; Pratt et al. 2007, for XMMNewton results; Ehlert & Ulmer 2009, for Chandra results; Sato et al. 2009, for Suzaku results) and numerical simulations (see Borgani et al. 2008, for a review).
Spatially resolved studies of the ICM temperature distribution in galaxy clusters are crucial for both astrophysical and cosmological applications since these studies provide information about the thermodynamic state of the ICM, allowing us to measure the total mass of these systems by means of the hydrostatic equilibrium equation (see e.g. Arnaud 2005; Colafrancesco & Giordano 2007) and to place constraints on the cosmological parameters by studying the cluster population evolution.
On the cosmological side, once the gas mass is determined from Xray observations in the deprojection or fitting analysis (see, e.g., Fabian et al. 1981; Ettori & Fabian 1999) and the total mass is estimated by means of the hydrostatic equilibrium condition, it is possible to derive the cluster gas mass fraction that can be used to constrain the cosmological density parameter, if combined with primordial nucleosynthesis calculations (e.g., White et al. 1993; Ettori et al. 2009). Cosmological applications of the cluster population evolution require precise measurements of the cluster temperature and of the correlation between temperature and other relevant physical quantities, to describe correctly the cluster population (total mass, luminosity, Compton parameter, number counts, redshift distribution, luminosity function). If well calibrated, the slope and evolution of cluster scaling relations, such as gas mass versus temperature (e.g., Colafrancesco & Vittorio 1994; Colafrancesco et al. 1994, 1997; Voit 2000; Finoguenov et al. 2001) and cluster size versus temperature (e.g., Colafrancesco et al. 1997; Mohr et al. 2000; Verde et al. 2001), can also be used to constrain cosmological and structure formation models.
On the astrophysical side, the precise determination of important quantities such as entropy, pressure, and  based on the assumption of hydrostatic equilibrium  the total cluster mass, is dependent on an accurate estimation of the temperature profile. Accurate temperature profiles are fundamental in determining the gas entropy distribution (see LloydDavies et al. 2000, and references therein), which is a powerful tool for exploring the role of nongravitational processes that could alter the specific thermal energy in the ICM (Kang et al. 2007; Colafrancesco & Giordano 2006, 2007; for a review see Colafrancesco 2008a), assessing the structure of cool cores and their surrounding atmosphere to probe the interplay of nonthermal and thermal particle distributions in clusters (e.g., Colafrancesco et al. 2004; Guo & Ho 2008; Colafrancesco & Marchegiani 2008, and references therein), and assessing the amount of nonthermal (pointlike and/or diffuse) emission mechanisms detectable in the hard Xray domain (see discussion in Colafrancesco & Marchegiani 2009).
The standard methodology for recovering the ICM temperature distribution uses high resolution spectroscopic Xray observations (XMM, Chandra, Suzaku and previously ROSAT, ASCA, and BeppoSAX). These observations provide the key measurable characteristics of the ICM, i.e., the temperature and density of this plasma. Because of limited photon statistics, the density and temperature are usually measured in terms of radial profiles (see, e.g., Pratt et al. 2007). However, while the density of the ICM is relatively easy to measure from the surface brightness profile of a galaxy cluster, precise temperature determination requires high photon statistics to build, and fit, a spectrum. Therefore, ICM temperature profiles are typically determined with considerably lower spatial resolution than density profiles. The measurement of radial temperature profiles is complicated by the density squared dependence of the thermal bremsstrahlung Xray emissivity, . The steep drop in the Xray surface brightness with distance from the centre, combined with the background from cosmic, solar, and instrumental sources, makes, hence, accurate Xray measurement of the temperature distribution at large distances from the cluster centre a technically challenging task (see discussion in Pratt et al. 2007).
The ICM temperature can alternatively be measured by using the
inverse Compton scattering of CMB photons off thermal electrons
residing in the cluster atmosphere  the SunyaevZel'dovich
effect (SZE; see Sunyaev & Zel'dovich 1980; Birkinshaw 1999;
Colafrancesco 2007, for reviews). This effect provides a CMB
temperature change
that depends on the cluster temperature directly from its amplitude (where is the Thomson crosssection, is the electron mass, is the electron density, is the cluster temperature, the integral is performed along the line of sight ), and from its spectrum g(x), where is the frequency normalized to the CMB energy, h is the Planck constant, is the Boltzmann constant, and T_{0} is the presentday CMB temperature. The temperature dependence is explicitly included in the function g(x) in terms of the relativistic effects of the Compton scattering that are more prominent in the high frequency region of the SZE spectrum at GHz (see Birkinshaw 1999; Colafrancesco et al. 2003; Colafrancesco 2007, for a review), where it is therefore possible to measure directly the ICM temperature. This is evident looking at the different temperature dependences of the three spectral characteristics of the thermal SZE signal:
 i)
 the minimum in its intensity located at the frequency
where , whose value depends weakly on the electron spectrum (i.e. on and ) and equals 2.26;  ii)
 the crossover frequency, x_{0}, whose value depends on the
electron pressure/energy density and optical depth
where and , and is found at a frequency > 3.83 for increasing values of (the value is found in the nonrelativistic limit, or in the limit );  iii)
 the maximum of its intensity located at the frequency
which depends sensitively on the electron population and its energy (momentum) spectrum (see Dolgov et al. 2001; Colafrancesco et al. 2003, 2009, for the case of electrons with a thermal spectrum; see also Colafrancesco et al. 2003; and Colafrancesco 2004, 2005, 2007, 2008b, for the case of electrons with different spectra).
The linear dependence of the SZE temperature change on the electronic density and temperature (see Eq. (1)) makes it possible  in addition  to enhance the sensitivity of the SZE measurements in the outer regions of the cluster while maintaining adequate sensitivity in the inner parts. Therefore, SZE observations could provide a more uniform and spatially extended coverage of the temperature profile in galaxy clusters than Xray observations.
Almost nothing is known about cluster temperature profiles obtained directly from SZE observations. Presentday and planned SZE observations from both groundbased experiments (multiband bolometers and interferometers) or upcoming spaceborne experiments (e.g., OLIMPO and Planck) require complementary Xray observations to place tight constraints on the cluster temperature, even though a somewhat more moderate precision on cluster temperature seems to be sufficient to achieve a reasonable statistical knowledge about the CDM power spectrum, in particular (Juin et al. 2007).
In this context, we emphasize that the use of Xray observed quantities in the SZE data analysis is, however, a delicate procedure, because the SZEderived temperature is weighted by the Compton parameter y. For instance, it is known that the derived peculiar velocity is systematically shifted by for this reason (see e.g., Hansen 2004).
While observations of the SZE are becoming increasingly accurate (e.g., De Petris et al. 2002; Battistelli et al. 2003; Halverson et al. 2009; Staniszewski et al. 2009), most analyses of SZ data are still based on the simplifying assumption of isothermality, or a single temperature plasma. Several studies have also considered the ability of future SZ observations to extract the cluster physical parameters (see, e.g., Knox et al. 2004; Aghanim et al. 2005), but always based on the assumption of isothermality.
A desirable solution with the SZE observations is to achieve simultaneous good angular resolution and high sensitivity to deproject the SZE signal properly throughout the entire cluster atmosphere.
We aim to develop an operational procedure to extract the fundamental parameters of the electron populations that produce the SZE, in both thermal and nonthermal plasmas. By applying this procedure, we identify the main instrumental characteristics required to achieve the scientific objectives of the SZE study that we propose to perform.
We discuss a strategy that is more effective in obtaining spatiallyresolved temperature measurements based only on SZE observations. This strategy relies on the following data being available: i) spatially resolved spectral measurements of the cluster temperature; ii) measurements of the temperature profile over a wide spatial range to differentiate the cool core region from the outer cluster atmosphere (or regions of different temperature in the overall atmosphere); iii) detailed and independent temperature measurements of wide frequency band coverage and sufficient spectral resolution.
With the same strategy, we also determine whether we can extract the basic parameters of a nonthermal plasma producing a nonthermal SZE that might add to or differ from the thermal SZE produced in a galaxy cluster.
The outline of the paper is as follows. We describe the general theory of the SZE in Sect. 2, and we generalize this theory to the case of clusters with nonuniform Tprofiles, such as cool core clusters, in Sect. 3. We describe a fitting procedure for spatially resolved spectroscopic SZE observations in Sect. 4, where we first discuss the case of the thermal SZE in the reference case of the A2199 cluster. The analysis of the nonthermal SZE and the combination of thermal and nonthermal SZE are also discussed, for completeness, in Sect. 5. The nonthermal SZE is important both for the intrinsic study of nonthermal and/or relativistic plasmas in clusters and in determining their role as biases in the study of the thermal plasma (i.e. the ICM). We present in Sect. 6 the application of our study for two specific galaxy clusters: Perseus, which has a cool core, and Ophiuchus, which is approximately isothermal. We discuss our results and summarize our conclusions in the final Sect. 7.
Throughout the paper, we use a flat, vacuumdominated cosmological model with , , and H_{0} = 70 km s^{1} Mpc^{1}.
2 The SZE: theory
To derive a SZE expression that is applicable to a galaxy cluster with spatially varying density and temperature, we need to recall the general SZE formulae.
We use the formalism presented by Wright (1979), and developed subsequently by Birkinshaw (1999), Enßlin & Kaiser (2000), and Colafrancesco et al. (2003).
We note that Bhm & Lavalle (2009) proposed a covariant formalism for the SZE, claiming, erroneously in the first version of their study, that the formalism presented in Wright (1979) and developed by other authors was incorrect. We verified (both numerically and analytically) that the two approaches, in the Thomson limit, are fully equivalent and provide the same results (see Appendix A for details). In addition to our verification, Nozawa & Kohyama (2009) also showed analytically that the two approaches are fully equivalent, pointing out the errors made by Bhm & Lavalle (2009).
The analytic expression for the upscattered CMB spectrum is
where
with (see e.g. Colafrancesco et al. 2003).
The function P(s) is given by the following expression
(7) 
where
is the electron optical depth and
where the symbol indicates the convolution product, and
The function is the electron momentum distribution function (with ) and is normalized to ensure that . The function P_{s}(s,p) is known from the basic physics of the Compton scattering and is given by
(11) 
where
Our aim is to invert Eq. (5) and derive the spectral and spatial parameters of the electron distribution from measurements of the SZE only. We derive an analytical solution to this problem, while in Sect. 4 we present a numerical procedure.
The expression for
given by a first order approximation in
is
(13) 
where
(14) 
Thus, by using Eq. (10), we can write
where
(16) 
By introducing the definitions
(17)  
(18) 
the previous Eq. (15) is given by
(19) 
whose formal solution is given by
with
(21) 
where is the identity operator.
3 The SZE for a galaxy cluster with a radial temperature profile
To apply the formalism described in Sect. 2 to a cluster with a spatially varying temperature and density profiles (i.e., a cluster with a cool core or a nonisothermal cluster), we must consider that:
 Eqs. (5)(9) still hold. In particular, the expression for the optical depth remains the same because it depends only on the gas density profile and not on the cluster temperature;
 the expression for the function P(s), and its series expansion in terms of powers of , remains formally unchanged, because it determines the probability that a CMB photon experiences a (logarithmic) change in the frequency after traversing the whole cluster;
 the expression for the function P_{1}(s)(i.e. Eq. (10)) must be changed. The expression given in Eq. (10) assumes that the electron momentum distribution function is the same throughout the cluster atmosphere. This assumption is incorrect for a cluster with a cool core (or a nonisothermal cluster) since its temperature, and hence the electron momentum distribution, changes with the cluster radius.
By describing the thermal electron density as
(22) 
it is possible to derive the average densityweighted momentum distribution along the line of sight
where
and and is the modified Bessel function of second kind (see, e.g., Abramowitz & Stegun 1965), which ensures the correct normalization such that
The function also satisfies the normalization given in Eq. (25), as can be verified by the equation
=  
=  (26) 
where we use Eq. (25).
We can, therefore, rewrite Eq. (10) for a cluster with a
generic temperature distribution by using the averaged electron
momentum distribution
where is given by Eq. (23). This function allows us to calculate the frequency redistribution probability for multiple scattering and, hence, the analytic expression of the SZE up to the required approximation order by using Eqs. (5)(9).
4 Fitting procedure: the thermal SZE
Equation (20) cannot be easily solved by analytical methods. We therefore describe a numerical procedure that can derive the cluster temperature profile from observations of the SZE of appropriate spatial and spectral resolution. We consider an instrument that is able to provide high quality SZE spectra in the frequency range 100500 GHz obtained with a spatial resolution of arcmin or less. We discuss the feasibility of such spatiallyresolved spectral observations in Sect. 7.
The quantitative analysis that we present in the following refers to the specific case of the cluster A2199 that has a cool core.
We simulate the thermal SZE observation of the cluster A2199 with parameters (density and temperature) taken from the available Xray observations, and extract the best fit values of the ICM temperature and optical depth from the simulated SZE observations. We consider, for convenience, the firstorder approximation to the thermal SZE, and we discuss in Sect. 5 the consequences of using this approximation. The temperature profile obtained from SZE measurements taken at various cluster radii is then deprojected based on the constraint (prior) that the density profile derived from SZE observations is consistent with that inferred from Xray measurements; we use this procedure, which assumes the Xray density prior, to ensure a fast enough computing time and allow us to explore the cluster in detail. We note, however, that the density constraint is not necessary, in principle, because the cluster density profile can be obtained, selfconsistently, from the optical depth profile measured from SZE observations. This fully consistent procedure requires, however, much longer computing times since the number of free parameters to be fitted to the observations is larger.
Figure 1: The SZE spectrum obtained for the cluster A2199 at different projected radii is shown with the frequency sampling derived, for each curve, with a 0.1% uncertainty. From top to bottom, curves refer to projected radii of 0, 5, 10, 15, 20, 30, 50, and 100 kpc. Note that spectra for 10 and 15 kpc are almost superimposed because beyond this radial distance the IC gas temperature measured by Chandra decreases with increasing radius (Johnstone et al. 2002). 

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Figure 2: The radial profile of the projected temperature ( top panel) and optical depth ( bottom panel) as derived from the fit to the SZE spectra using data sampled in the frequency range 300400 GHz with uncertainty of 0.1% (see Fig. 1). 

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For the reference case of A2199 that we now discuss,
our analysis included the following steps:
i) we considered Xray information on A2199 available from Chandra
observations (Johnstone et al. 2002);
ii) using the density and temperature profiles derived from Xray
observations, we calculated (using the procedure described in
Sect. 3 above) the SZE spectrum at eight different projected radii
from the cluster center, i.e.,
,
5, 10, 15, 20, 30,
50, and 100 kpc, where 1 kpc corresponds to 1.67 arcsec at the
redshift z = 0.03 of A2199;
iii) for each radial bin, we sampled the calculated SZE
spectrum at six different frequencies in the range 300400 GHz,
which is the most sensitive part of the SZE spectrum to the cluster
temperature, i.e.
300, 320, 340, 360, 380 and 400 GHz (see
Fig. 1); iv) for each radius, we fitted the six frequency SZE data with a
relativistic model of the thermal SZE leaving free two parameters,
i.e.
and ,
and assuming for each experimental data
point an uncertainty of
(we also consider below the
case of 1% data uncertainty, for comparison). For each projected
radius, this fitting procedure yields a value of the
projected temperature and optical depth (see Fig. 2);
v) to deproject the fitted temperature profile, we
assumed a 3D theoretical temperature profile of the form
(see e.g. Piffaretti et al. 2005). To decrease the number of free parameters in the fit we assumed the ICM density profile obtained from Xray data. In the fitting procedure, the projected temperature is calculated by weighting the temperature profile by the gas density and integrating along the line of sight to obtain
where R is the cluster radius (and it is R=200 kpc for the case of A2199). Assuming the value of from Xray observations, the number of free parameters reduces to three. In this way, it is possible to fit the projected temperature obtained from the simulated SZE observations, and then derive the temperature profile parameter (in Eq. (28)); assuming keV for the case of A2199, the bestfit model parameters are keV, kpc, and . The bestfit temperature profile curve is shown in Fig. 3, where it is compared to the deprojected temperature data derived from Xray observations.
Figure 3: The deprojected temperature profile curve (solid) derived from SZE spectral data is compared to the deprojected temperature data obtained from Xray observations of A2199 (Johnstone et al. 2002). 

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The previous fitting procedure was repeated assuming that the SZE observations sampled from Fig. 1 have an uncertainty of . The radial profile of the projected temperature obtained in this case is shown in Fig. 4. In this case, the deprojection of the temperature profile using the template given in Eq. (28) does not provide an acceptable fit. We therefore conclude that an uncertainty of order of in the SZE data is too large to derive a detailed profile of the ICM temperature in clusters like A2199.
Figure 4: The projected temperature profile obtained from the fit to SZE spectra using data points sampled in the 300400 GHz range and assuming an uncertainty of 1%. 

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The entire fitting procedure was repeated by sampling the SZE observations in the lowfrequency range 100200 GHz (, 120, 140, 160, 180, 200 GHz) in which the SZE spectrum is, however, much less sensitive to the cluster temperature. In this case, we find that, even assuming uncertainties of 0.1%, it is impossible to reproduce satisfactorily the radial profile of the projected temperature derived from the fit by using the template in Eq. (28) (see Fig. 5). This is because  as already emphasized  the shape of the SZE spectrum at low frequencies is less sensitive to the cluster temperature than the shape of the highfrequency part of the SZE spectrum. We therefore conclude that, to obtain detailed information about the cluster temperature profile, one must use highfrequency spectral observations of the SZE in the range 300400 GHz. This is crucial when designing the strategy behind SZE observations with the goal of deriving strong constraints on the fundamental parameters of the cluster.
Figure 5: The projected temperature radial profile obtained from the fit to the SZE spectral data by using data points sampled in 100200 GHz range and assuming uncertainties of 0.1%. 

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In the frequency range 200300 GHz, which contains the null of the thermal SZE, the kinematic and/or the nonthermal SZE might produce significant additional contributions to the overall SZE. Therefore, in this frequency range, the analysis that we have performed in this section should be performed at a later stage, only after we have estimated the main parameters of the cluster gas, and , by using the SZE measurements in the highfrequency range, 300400 GHz (see Sect. 5.2), which are much less sensitive to the biases caused by the superposition of the kinematic and nonthermal SZE.
5 Fitting procedure: the nonthermal SZE
We simulate the fitting procedure of a nonthermal SZE produced in an extended source such as a cluster radio halo or the lobes of a radio galaxy (see e.g. Colafrancesco 2008) and we consider the effects of the variation in the parameters of the nonthermal electron spectrum on the relative nonthermal SZE.
We first study the spectral shape of the nonthermal SZE as a
function of the electron spectral parameters. We
use an arbitrary normalization in which the amplitude of the SZE
at its minimum is set to be equal to 1, and we vary the
parameters of the electron spectrum. Figure 6 shows
the SZE spectrum for a single powerlaw spectrum
(30) 
(here is the normalized momentum), where s_{1}=3.0, as a function of the minimum momentum p_{1}. It is clear that the frequency position of the minimum of the SZE for high values of p_{1} is found around 160 GHz and does not change for values p_{1}> 5, while it is considerably lower (of order of 140 GHz) only for low values of p_{1}, such as . The shape of the spectrum at higher frequencies ( GHz) is, instead, more sensitive to the value of p_{1}.
Figure 6: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the minimum momentum p_{1} of the electron spectrum. We assume here a single powerlaw spectrum with powerlaw index s_{1}=3.0. 

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We show in Fig. 7 the case of a double
powerlaw spectrum
where s_{1}=0.1, , and s_{2}=3.0. In this case, the position of the minimum and the shape of the spectrum at frequencies GHz, depends weakly on the value of p_{1}, while a stronger dependence from p_{1} is present at higher frequencies (this dependence is weaker than in the single powerlaw case because of the lower power at the lowp end of the electron spectrum).
Figure 7: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the minimum momentum p_{1} of the electron spectrum. We assume here a double powerlaw spectrum with s_{1}=0.1, s=3.0, and . Spectral curves are for p_{1}=1 (solid), 10 (dashed), and 100 (dotdashed). 

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Figure 8: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the spectral index s_{1} of the electron spectrum. We assume here a single powerlaw spectrum with p_{1}=1. Curves are for s_{1}= 3 (solid), 2.5 (dashed), and 3.5 (dotdashed). 

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Figure 8 shows the changes in the SZE spectrum when we consider a single powerlaw with p_{1}=1 as a function of the spectral index s_{1}: the shape of the SZE spectrum is more sensitive to the shape of the electron spectrum at frequency GHz. Figure 9 shows the variation in the SZE spectrum as a function of the spectral index s_{2} for a double powerlaw spectrum: the dependence on the electron spectrum index is quite weak for the whole frequency range considered.
Figure 9: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the spectral index s_{2} of the electron spectrum. We assume here a double powerlaw spectrum with p_{1}=1, s_{1}=0.1 and . 

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After illustrating the dependence of the SZE spectrum on the electron spectrum parameters, we discuss the procedures by which we can recover the physical parameters of the nonthermal electron spectrum from the SZE observations. We consider here two cases: i) a pure nonthermal electron population with a powerlaw spectrum (a situation applicable to radio galaxy lobes, see e.g. Colafrancesco 2008b); ii) a combination of thermal plus nonthermal electron population with a double powerlaw spectrum (a situation applicable to galaxy clusters with nonthermal phenomena, see e.g. Colafrancesco et al. 2003).
5.1 The case of a single powerlaw spectrum
We describe how to derive the parameters of an electron population with a single powerlaw spectrum, assuming that this is the population principally responsible for the SZE production.
Following the procedure previously outlined for the thermal
SZE in Sect. 4, we sampled the calculated nonthermal SZE
produced at different projected radii and at six different
frequencies in the range 300400 GHz, where, as previously
noticed, the shape of the SZE is most sensitive to the
physical parameters of the electron spectrum.
The nonthermal emission region was assumed, for simplicity,
to have spherical symmetry, a radius of 50 kpc, and an
electron spectrum
where we assume s_{1}=3.0, p_{1}=3, k_{0}=10^{2} cm^{3}, and a radial distribution of the electron population given by
with kpc and . These parameters are typical of giant radio galaxy lobes.
In the fitting procedure, we fixed the spectral index of the electron spectrum s_{1}, because it can be derived from radio observations, and we fit the simulated observations with two free parameters: the optical depth and the minimum momentum p_{1}. We note that we did not need to assume the parameter s_{1}derived from radio data, because, with adequate frequency coverage, it can be derived from the SZE data alone; this assumption is made for the sole purpose of reducing the numerical computation time.
Figure 10: The nonthermal SZE spectrum evaluated at various projected radii and frequency sampling (see text for details) with data uncertainty of 0.1%. Curves are, from bottom to top, for projected radii of 0, 5, 10, 15, 20, 30, 40 kpc. 

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Figure 11: The radial profile of the optical depth as obtained from the fit to the SZE spectra b using data sampled in the range 300400 GHz with uncertainty of 0.1% (see Fig. 10). 

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The SZE intensity at different radii and the relative sampling are shown in Fig. 10. The behaviour of the optical depth at different projected radii, as derived by the fitting procedure, is shown in Fig. 11. Our analysis recovers the input value (see Eq. (32)) at all considered projected radii. To deproject the optical depth radial distribution and derive the radial profile of the electron density, it is sufficient to fit the optical depth profile with the density profile given in Eq. (33) and integrate along the line of sight. The deprojection procedure yields cm^{3}, kpc, and , which are consistent, within the errors, with the input values that we used in the simulations.
We emphasize that the previous analysis of the SZE spectral measurements allowed us to derive directly the density profile and the spectral shape of the electron distribution in the radio galaxy lobes. Therefore, the combination of SZE spectral measurements and the measurement of the lowfrequency radio emission allows us to break the degeneracy affecting synchrotron emission between the electron density and the magnetic field and eventually yield a precise measure of the magnetic field in radio galaxy lobes (see also Colafrancesco 2008b).
5.2 The case of thermal plus nonthermal populations
We consider a combination of a thermal electron distribution that reproduces the A2199 cluster data and an additional nonthermal electron spectrum with a double powerlaw spectrum, as in Eq. (31), with parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and . For the radial (thermal and nonthermal) components, we consider the density profile given in Eq. (33) with parameters kpc and . These parameters accurately reproduce the density profile of A2199 obtained from Xray observations (see discussion in Sect.4).
A nonthermal electron spectrum similar to a double power law is expected in galaxy clusters because the nonthermal electrons of relatively low energy lose energy by means of Coulomb interactions with the thermal IC gas, and these energy losses flatten the electron spectrum at low (150 MeV) energies (see, e.g., Colafrancesco & Marchegiani 2008, 2009, for a discussion). We do not follow a strategy identical to that described in previous sections, in that we do not perform a fit to the sampled points because this procedure would involve too many free parameters (i.e., those of both the thermal and the nonthermal electron populations). We instead focus on the spectral regions where the nonthermal SZE is expected to be more relevant and therefore can be measured. For this purpose, we compare the nonthermal SZE spectrum with both the thermal SZE spectrum, calculated both at first order in and with second order corrections (see Sect. 2 and Colafrancesco et al. 2003, for more details), and the spectrum of a possible SZ kinematic effect. The spectral regions where the nonthermal SZE could be most reliably estimated are i) the frequency range around the minimum of the SZE (at GHz), because all amplitudes are basically proportional to the optical depth of the relative electronic population; and ii) the frequency range around the crossover frequency of the thermal SZE (at GHz), because the thermal SZE is by definition null at this frequency, leaving hence visible the (negative in sign) nonthermal SZE.
We note that an upper limit to the nonthermal electron density can be derived, after fixing their spectral shape, by requiring that the nonthermal pressure does not exceed the thermal one (see discussion in Colafrancesco et al. 2003). Figure 12 shows the ratio of the nonthermal SZE to the thermal SZE in the case at the cluster center: this case can be considered as an upper limit to the contribution of the nonthermal component to the total SZE. Figure 13 shows the spectrum of the thermal and thermal plus nonthermal SZE in the frequency ranges 100200 GHz and 300400 GHz. The ratio of the nonthermal to the thermal SZE is large only in the region around GHz, where the thermal SZE has its crossover and where, consequently, the total SZE is dominated by the nonthermal component. For GHz, the nonthermal SZE is of the thermal one, while for GHz, the ratio becomes very low, of order of or even less. Two conclusions can be derived from these results: 1) in the frequency range 300400 GHz, which is optimal for deriving precise information about the thermal SZE, the nonthermal component does not provide a contribution that jeopardizes the precision of the measurements; 2) at lower frequencies, around 200 GHz, the contribution of the nonthermal component is more relevant but, to measure this component, we must differentiate it from the thermal one.
A possible observational means of separating the nonthermal effect from the thermal one consists of measuring the thermal SZE in the frequency range 300400 GHz and deriving the IC gas physical parameters, as shown in Sect. 4. Then, once the thermal SZE parameters are fixed, it is possible to measure the SZE in the frequency range 100200 GHz and derive the parameters of the nonthermal population that, in this frequency range, produce a contribution to the total SZE of order of (see Figs. 12 and 13): such a SZE amplitude is measurable if measurements have an uncertainty no greater than .
Figure 12: The frequency dependence of the ratio of the nonthermal SZE (we assume a double powerlaw spectrum with s_{1}=0.1, s_{2}=3.0, p_{1}=1, and ) to the thermal SZE at the center of A2199. We assume a pressure balance of at the cluster center. 

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Figure 13: The thermal (solid curve) and thermal plus nonthermal (dashed curve) SZE evaluated at the center of A2199. The nonthermal electron population has a double powerlaw spectrum with parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and and a pressure at the cluster center. The spectra are shown in the frequency range 100200 GHz ( upper panel) and 300400 GHz ( lower panel). 

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To precisely determine the cluster parameters, we must take into account the contribution of the thermal SZE evaluated at second order in , which can be calculated in detail once the main parameters of the IC gas are known (see Colafrancesco et al. 2003, for details). The modifications induced by the second order effect, compared to both the nonthermal SZE and the firstorder thermal SZE, are shown in Fig. 14. The frequency range 300400 GHz is the optimal frequency window for extracting the thermal SZE parameters, and its secondorder correction to the thermal SZE is less than 30% of the nonthermal SZE and less than 0.1% of the firstorder thermal SZE. Hence, we can conclude that the total SZE can be adequately approximated, in this frequency range, only by the firstorder thermal SZE. In the frequency range 100200 GHz, where the nonthermal effect can be measured, the second order correction to the thermal effect is of the nonthermal effect, so that the second order correction can also be neglected for this purpose.
Figure 14: Upper panel: the frequency dependence of the absolute value of the ratio of the second order correction to the thermal SZE at the center of A2199 to the nonthermal SZE of an electron population that has a double powerlaw spectrum with parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and and pressure . Lower panel: the frequency dependence of the ratio of the second order correction to the first order description of the thermal SZE. 

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We also consider now the possible contribution of a kinematic SZE, whose amplitude is maximal at frequencies close to the crossover frequency of the thermal SZE, i.e. at GHz. Figure 15 shows the ratio of the kinematical to the nonthermal SZE (upper panel) and the ratio of the kinematical to the thermal SZE (lower panel) for a receding cluster with peculiar velocity of 1000 km s^{1}. In the frequency range 300400 GHz, the result is that the kinematical contribution is a small fraction () of the nonthermal SZE, and less than 0.1% of the thermal one. Therefore, the kinematical SZE can also be neglected in this frequency range. Figure 15 also shows that in the range 100200 GHz the kinematical SZE is less than 10% of the nonthermal SZE. However, if the nonthermal electron population were to have a pressure , then the nonthermal and kinematical SZE would be comparable (based on the assumption of km s^{1}); in this case, however, these signals would be indistinguishable from the thermal one (for reference see Figs. 12 and 13 upper panel, in which we show the case ).
Figure 15: Upper panel: the frequency dependence of the absolute value of the ratio of the kinematic SZE for a value of the peculiar velocity of km s^{1} to the nonthermal SZE for an electron population with the same spectrum as in Fig. 14. Lower panel: the frequency dependence of the absolute value of the ratio of the kinematic SZE to the thermal SZE evaluated to first order. 

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Our results can be summarized as follows:
 i)
 in the frequency range 300400 GHz, the first order thermal SZE is a good approximation of the total SZE (all the corrections are ).
 ii)
 in the frequency range 100200 GHz, if the nonthermal contribution is strong, it dominates over the kinematic one. If, instead, the two SZE contributions are comparable, then they cannot be easily differentiated from the thermal one. Hence, if deviations from expected thermal SZE are measured in this frequency range, they should be of nonthermal origin;
 iii)
 at GHz (i.e. the crossover frequency of thermal SZE) alone, the kinematic SZE, the nonthermal SZE and the second order thermal correction are important with respect to the first order thermal effect. To differentiate between the three different contributions, it is important to place constraints on the thermal and the nonthermal parameters by performing precise observations in frequency ranges 300400 and 100200 GHz, respectively.
6 Applications to specific clusters: Perseus and Ophiuchus
We present our studies of two nearby and extended clusters: Perseus, which has a cool core, and Ophiuchus, which is approximately isothermal. As discussed previously, the optimal strategy for extracting the physical parameters of the cluster is to consider the frequency range 300400 GHz; in this range, we can safely consider only the first order thermal SZE in our analysis, because higherorder corrections are negligible.
6.1 The Perseus cluster
The Perseus cluster is at a redshift z=0.0179 or a distance of 77.7 Mpc, at which 1 arcmin corresponds to 21.8 kpc. Therefore, a study of the spatially resolved SZE allows us to achieve good spatial sampling of the temperature profile, superior to that of the cluster A2199 previously discussed. In the following, we repeat our analysis for this cluster.
Churazov et al. (2003) derived an analytical approximation to the radial profile of the IC gas density and temperature that reproduces the cluster data for r>10 kpc, while at shorter distances the central dominating galaxy NGC 1275 causes the IC gas radial density profile to be more uncertain. For the sake of illustration, we use the analytical fitting formulae given by Churazov et al. (2003) and estrapolate them to the cluster center, ignoring, in this way, the presence of the central galaxy.
Figure 16: The thermal SZE spectra of the Perseus cluster evaluated at different projected radii of 0, 20, 50, 75, 100, 150, and 200 kpc ( from top to bottom) are shown in the frequency range 200500 GHz. For each curve, the SZE sampling in the frequency range 300450 GHz has an uncertainty of 1%. 

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Figure 17: The radial profile of the projected temperature, derived from the fit to the SZE spectra in Perseus in the frequency range 300440 GHz with uncertainties of 1% (see Fig. 16), is shown as a function of the projected radius (in arcmin). 

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Figure 18: The deprojected temperature radial profile in Perseus derived from the fit to the Xray data by Chrurazov et al. (2003) (solid curve) is compared to the fit to the SZE simulated data (dashed curve) shown in Fig. 17. For the SZE temperature profile, we use the radial profile given by Eq. (28), with keV, keV, kpc, and . 

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Figure 19: As Fig. 17 but with uncertainties assumed at the level of 0.1% of the sampled SZE data points (see Fig. 16). 

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We hence simulated to first order the thermal SZE expected for Perseus at seven projected radii from the cluster center: 0, 20, 50, 75, 100, 150, and 200 kpc (corresponding to angular sizes of 0, 0.9, 2.3, 3.4, 4.7, 6.9, and 9.2 arcmin). We then sampled each one of the SZE spectra at these radii for eight different frequencies in the 300450 GHz range: 300, 320, 340, 360, 380, 400, 420, and 440 GHz (see Fig. 16). We extended here the frequency coverage of the SZE spectra to improve our sampling of the spectral region of the SZE where relativistic effects are relevant. For each projected radius we fitted, using the gas temperature and optical depth as free parameters, the SZE simulated signal by using the expression of the firstorder thermal SZE and assuming an error of 1% in the experimental data. The results of the fit to the projected temperature are shown in Fig. 17. The deprojection procedure of the bestfit radial temperature profile is performed by using Eq. (28), defining keV (the bestfit projected temperature at 200 kpc), and by using as prior constraints the information about the electron radial profile provided by Xray observations (see Sect. 4). For an experimental uncertainty of 1%, such procedure yields values keV, kpc, , where for 4 d.o.f. Figure 18 compares the deprojected temperature profile found with our method from SZE observations with the bestfit deprojected temperature profile derived by Churazov et al. (2003) obtained from Xray data. The two curves are quite similar except for the inner region, where the IC gas is cooler and the sensitivity of the SZE spectrum to low temperatures (because of relativistic corrections) is lower. We also verified that by assuming an uncertainty of 0.1% in the experimental data, the bestfit values do not change, but the relative errors are reduced considerably (see Fig. 19).
6.2 The Ophiuchus cluster
To first approximation, the Ophiuchus cluster is an isothermal cluster with keV (Watanabe et al. 2001). Since Ophiuchus is quite a hot cluster, relativistic effects in the SZE spectrum are expected to be more prominent, allowing us to obtain a more robust estimate of both the cluster gas temperature and density.
The SZE signal produced in Ophiuchus, calculated as described in Sect. 2, was sampled at six frequencies between 300 and 400 GHz and at seven different radii, 0, 35, 70, 120, 200, 300, and 500 kpc (see Fig. 20), assuming experimental uncertainties of 1%. The minimum radius at which we sampled the radial profiles was chosen by taking into account that at the Ophiuchus distance (z=0.028), 1 arcmin corresponds to 34 kpc.
Figure 20: The thermal SZE spectrum for the Ophiuchus cluster is shown at various projected radii of 0, 35, 70, 120, 200, 300 and 500 kpc ( from top to bottom). The assumed uncertainty in the SZE data points is 1%. 

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Figure 21: The radial profile of the projected temperature ( upper panel) and of the IC gas density ( lower panel) as derived from the fit to the SZE spectra for Ophiuchus shown as a function of the projected radius (in arcmin). We use SZE data points in the frequency range 300400 GHz with uncertainties of 1% (see Fig. 20). 

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Figure 21 shows the radial profiles of the
projected temperature and optical depth of the IC gas obtained
from the fit to SZE data. These results clearly indicate that
the cluster temperature is constant with radius, within the
errors, while the radial density profile can be derived from the
optical depth radial profile by using the model
We therefore note that in a cluster with constant temperature profile, deprojection can be performed using only optical depth results without assuming any prior constraint obtained from Xray data. Hence, the thermal gas parameters derived from SZE and Xray data can be directly compared.
The values derived from Xray measurements are cm^{3}, kpc, and (Johnston et al. 1981; Watanabe et al. 2001). The values that we derive from the fit to the simulated SZE signals are cm^{3}, kpc, and . The comparison between the bestfit profile to Xray data and that to the SZE data is shown in Fig. 22. We can see that the density profile derived from SZE measurements is very similar to that obtained from Xray observations.
Figure 22: The IC gas density profile of Ophiuchus derived from Xray observations (solid curve) compared with that derived from the optical depth radial profile shown in Fig. 21 (dashed line) obtained from SZE simulated observations (see text for details). 

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7 Discussion and conclusions
We have derived a general formalism to describe the thermal SZE in galaxy clusters with a nonuniform temperature profile, which can be applied to both coolcore clusters and noncool core cluster with isothermal or nonisothermal temperature structures.
To extract the temperature profile information from spatiallyresolved, spectroscopic SZE observations, we have developed an inversion technique by which the electron distribution function can be extracted using a SZE signal with a wide frequency range coverage. We have presented extensive results of the fitting procedure used to extract the cluster gas temperature from a set of simulated spatially resolved spectroscopic SZE observations in different bands of the spectrum, from 100 to 450 GHz, and we discuss the optimal experimental and theoretical strategy.
Our analysis of three different cluster prototypes (A2199 with a lowtemperature cool core, Perseus with a relatively hightemperature cool core, and Ophiuchus with an isothermal temperature distribution) inferred the required precision of the SZE observations and the optimal frequency bands to determine the cluster gas temperature with an accuracy similar to or better than that obtainable from Xray observations. The precision level of SZEderived temperature was also discussed for the outer regions of clusters, which are particularly relevant to accurate mass determination of galaxy clusters. We also investigated how to extract from our method the parameters characterizing the nonthermal SZE spectrum of the relativistic plasma contained in various astrophysical environments, such as the lobes of radio galaxies and the relativistic electrons cospatially distributed with the thermal plasma in galaxy clusters with nonthermal phenomena (e.g., radio halos and/or hard Xray excesses).
Based on our results, we conclude that the method we present here is the only method, so far, for SZE observations that is able to extract the crucial parameters of the cluster atmospheres (i.e. their temperature, density and additional nonthermal components) using only a single observational technique, i.e. spatially resolved spectroscopic SZE observations. These SZE observations are becoming available with the advent of new, but challenging, experimental techniques.
To achieve a wide spectral range (i.e. the 100500 GHz range discussed in our analysis) with imaging capabilities, a Fourier Transform spectrometer would be an optimal experimental solution. Even though FT spectrometers already operate on Herschel(Spire), specific technical solutions for spaceborne projects using MartinPuplett architecture (Martin & Puplett 1970) have been presented (see, e.g., DeBernardis et al. 2010). It has been shown in fact that these experimental solutions allow a clean differentiation of the spectra coming from two independent inputs covering two contiguous areas of the telescope focal plane, so that the measured spectrum is the difference in the spectra of the target source and an offset reference field (see, e.g., DeBernardis et al. 2010). For this differential configuration, even small SZE signals can be extracted from a strong mode background generated, e.g., by the CMB itself, and the intrinsic emission from the telescope (either radiatively cooled to 80 K, or actively cooled to a few K), in addition to other signals from the Earth and other astronomical foreground sources.
Fourier Transform spectrometer experiments have the crucial advantage (with respect to dispersion spectrometer experiments) of being imaging instruments, an essential requirement for the analysis we have presented in this paper. With this choice, hence, a large detector array can be arranged in the focal plane of a submm telescope, thus increasing the mapping speed.
To limit the radiative background (and the corresponding fluctuations) of the detectors, and match the detector size to the diffractionlimited angular resolution (which improves with frequency), the total spectral range of interest can be divided into different frequency bands. The frequency ranges of these bands must then be optimized to achieve optimal performance when studying the SZE.
The bands resulting from the SZE study optimization are B_{1} = 100200 GHz, B_{2} = 200300 GHz, and B_{3} = 300450 GHz. An additional fourth band, B_{4} = 600800 GHz, could be useful in controlling the spectral behaviour of the foreground emission.
The lowtomedium resolution configuration (i.e., a 10 to 20 GHz resolution, but constant over a large frequency band) for the spectrometer envisaged here is completely suitable to study the smooth features of the continuum SZE spectra. We note that the different spectral resolutions (from 20 to 1 GHz) achievable by Fourier Transform spectrometer enable a wide range of astrophysical sources to be studied.
With a 3 m diameter dish, the corresponding angular resolution, which is limited by diffraction at the telescope entrance aperture, of the previous three bands used to measure the SZE are approximately 4.5, 2.25, and 1.5 arcmin FWHM, respectively. With a 12 m diameter dish, the angular resolution can be reduced to approximately 1.06, 0.56 and 0.38 arcmin FWHM, respectively.
For an instrument field of view (FOV) of arcmin^{2} (resulting in two contiguous FOVs in the sky in the differential imaging spectrometer strategy) filled with diffraction limited detectors, the resulting number of detectors for each of the above frequency bands are approximately (D/4.5)^{2} [ (D/1.06)^{2}], (D/2.25)^{2} [ (D/0.56)^{2}], and (D/1.5)^{2} [ (D/0.38)^{2}] for the 3 m (12 m) diameter dish, respectively.
These optimal spectroscopic capabilities must be coupled to highsensitivity detectors to fully exploit the large amount of information enclosed in the SZE signal. The photon background depends on the cooling of the whole system: it is dominated by the warm telescope (80 K) for a passive cooling, but can be strongly reduced by using active cooling that could bring the telescope to a few K. The achievable photon noise limited NEP of each of the detectors can be in the range 10^{17} to 10^{19} for a passive or active cooling, respectively, therefore allowing full exploitation of even the smallest SZE signals.
To summarize, the analysis of the SZE signals that we have presented in this paper requires angular resolution of order of arcmin (or less), a spectral resolution of order of 10 GHz, and a sensitivity of order of 0.1 to 1% of the predicted SZE signals, which are achievable for integration times of minutes to hours with an experimental set up similar to the one envisaged here.
Other methods for extracting the cluster temperature profile by using SZE observations have been presented (Holder & Loeb 2004), but they use a combination of the thermal SZE (which measures the electron pressure distribution) and radio observations of the highly polarized scattered radiation coming from active galaxies associated with the cluster (which depends on the cluster electron density distribution). Holder & Loeb pointed out that current instruments should allow us to reach accuracies in our measurements of the massweighted cluster temperature profiles of order of 1 keV, but based on the assumption that the central radio source is stable over several million years. However, variable or beamed sources (such as radio galaxies or jets/lobes of AGNs associated with the cluster) will leave observable signatures in the scattered emission. Therefore, this method could allow, in principle (e.g. using K sensitivity observations of the polarized emission), the measurement of the age of the central source by finding an edge to the polarized emission.
In this context, we note that combining this method with the independent temperature measurements from SZE spectral studies (that we discuss here) would allow us to measure the time evolution of the central radio source.
In conclusion, our study shows that SZE experiments with spectroscopic capabilities, such as those using Fourier Trasform techniques with imaging capabilities, can measure precise temperature distribution over large radial distances for galaxy clusters out to high redshift. This will allow SZE observations of clusters to be used to improve our understanding of the physics of cluster atmospheres, to reconstruct their total mass content and, eventually, to use clusters of galaxies as reliable astrophysical and cosmological probes.
Appendix A: Comparing our results to the covariant formalism derivation of the SZE
We compare the SZE calculated according to the formalism of Wright (1979; W79) with that calculated by the derivation of Bhm & Lavalle (2009; BL09). For the second derivation, we refer to their Eqs. (18)(22), derived in the Thomson limit, which are therefore directly comparable to those derived by W79.
According to BL09, the correct expression of the SZE in the
relativistic covariant formulation, is
The averages over the angles of the two contributions are given by the expressions
(A.2) 
where in the Thomson limit, and
Here, the function is normalized such that , where , and the function is given by
where .
To quantitatively compare the two approaches, we considered
an electron population with a double powerlaw spectrum
where , normalized to ensure that . We assumed the parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and . By using the adimensional momentum p and the previous normalization, we note that the expression for the function in Eq. (A.4) slightly changes. For consistency, it must be multiplied by the factor , derived using Eqs. (16), (17), and (53) in BL09.
Figure A.1 plots the SZE spectrum over a wide frequency range, from x=10 ( GHz) to x=10^{15} ( GeV) calculated to first order for according to W79 (continuous curve), and that calculated according to BL09 (dashed line). As can be clearly recognized, the two spectra are indistinguishable over the entire frequency range.
Figure A.1: The SZE spectrum calculated following the method of W79 (solid line) and the method of BL09 (dashed line). The horizontal axis shows the adimensional frequency , while the vertical axis shows the SZE brightness change in units of . No difference between the two approaches is apparent. 

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Figure A.2 shows a zoomed view of the SZE spectra in the microwave region (x=0  20). In this region, the results obtained by the two formalisms are also indistinguishable.
Figure A.2: Same as Fig. A.1 but with a zoom in the microwave frequency range at which the SZE telescope operate. In this lowfrequency range, there is also no difference between the two approaches. 

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Figure A.3 compares the ICS emission (against the CMB photons) along the line of sight calculated in the approach of BL09 and in the standard approach (see e.g. Colafrancesco et al. 2005; see also Longair 1993) using also the KleinNishina crosssection, and assuming an electron spectrum as in Eq. (A.5), with a density and spatial distribution such that . The two different calculations provide quite similar results, apart for the frequency range x > 10^{15}, at which the KleinNishina crosssection has a nonnegligible effect (it indeed reduces the Compton scattering efficiency at high energies, see e.g. Fargion & Salis 1998).
Figure A.3: The comparison of the ICS emission evaluated following the standard computational approach with the inclusion of the KleinNishina crosssection (solid line) and that evaluated following the method of BL09 in the Thomson approximation (dashed line). No difference appears between the two approaches up to a frequency . 

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Based on these results, we can conclude that:
 i)
 For the SZE spectrum in the microwavemm region, the formalism of W79 can also be used in the case of relativistic electrons since it provides the same results as the covariant formalisms of BL09. In addition, the W79 formalism, which uses a simple expression, requires a much shorter computing time w.r.t. the BL09 formalism.
 ii)
 When calculating the ICS emission at high frequencies, the standard formalism is completely adequate because there is no difference w.r.t. the covariant formalism results. In addition, the standard formalism requires a much shorter computing time and allows also the KleinNishina crosssection effect to be used without introducing additional complications in the numerical computation, as occurs  in contrast  in Eq. (46) of BL09.
To summarize, the W79 and the covariant formalisms that have been developed to calculate the SZE are fully equivalent, in contrast to previous erroneous claims (see BL09), and the significant advantage of the standard calculations based on the W79 approach is their far shorter computing times.
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All Figures
Figure 1: The SZE spectrum obtained for the cluster A2199 at different projected radii is shown with the frequency sampling derived, for each curve, with a 0.1% uncertainty. From top to bottom, curves refer to projected radii of 0, 5, 10, 15, 20, 30, 50, and 100 kpc. Note that spectra for 10 and 15 kpc are almost superimposed because beyond this radial distance the IC gas temperature measured by Chandra decreases with increasing radius (Johnstone et al. 2002). 

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In the text 
Figure 2: The radial profile of the projected temperature ( top panel) and optical depth ( bottom panel) as derived from the fit to the SZE spectra using data sampled in the frequency range 300400 GHz with uncertainty of 0.1% (see Fig. 1). 

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In the text 
Figure 3: The deprojected temperature profile curve (solid) derived from SZE spectral data is compared to the deprojected temperature data obtained from Xray observations of A2199 (Johnstone et al. 2002). 

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In the text 
Figure 4: The projected temperature profile obtained from the fit to SZE spectra using data points sampled in the 300400 GHz range and assuming an uncertainty of 1%. 

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In the text 
Figure 5: The projected temperature radial profile obtained from the fit to the SZE spectral data by using data points sampled in 100200 GHz range and assuming uncertainties of 0.1%. 

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In the text 
Figure 6: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the minimum momentum p_{1} of the electron spectrum. We assume here a single powerlaw spectrum with powerlaw index s_{1}=3.0. 

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In the text 
Figure 7: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the minimum momentum p_{1} of the electron spectrum. We assume here a double powerlaw spectrum with s_{1}=0.1, s=3.0, and . Spectral curves are for p_{1}=1 (solid), 10 (dashed), and 100 (dotdashed). 

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In the text 
Figure 8: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the spectral index s_{1} of the electron spectrum. We assume here a single powerlaw spectrum with p_{1}=1. Curves are for s_{1}= 3 (solid), 2.5 (dashed), and 3.5 (dotdashed). 

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In the text 
Figure 9: The nonthermal SZE spectrum, normalized to its minimum amplitude is shown as a function of the spectral index s_{2} of the electron spectrum. We assume here a double powerlaw spectrum with p_{1}=1, s_{1}=0.1 and . 

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In the text 
Figure 10: The nonthermal SZE spectrum evaluated at various projected radii and frequency sampling (see text for details) with data uncertainty of 0.1%. Curves are, from bottom to top, for projected radii of 0, 5, 10, 15, 20, 30, 40 kpc. 

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In the text 
Figure 11: The radial profile of the optical depth as obtained from the fit to the SZE spectra b using data sampled in the range 300400 GHz with uncertainty of 0.1% (see Fig. 10). 

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In the text 
Figure 12: The frequency dependence of the ratio of the nonthermal SZE (we assume a double powerlaw spectrum with s_{1}=0.1, s_{2}=3.0, p_{1}=1, and ) to the thermal SZE at the center of A2199. We assume a pressure balance of at the cluster center. 

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In the text 
Figure 13: The thermal (solid curve) and thermal plus nonthermal (dashed curve) SZE evaluated at the center of A2199. The nonthermal electron population has a double powerlaw spectrum with parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and and a pressure at the cluster center. The spectra are shown in the frequency range 100200 GHz ( upper panel) and 300400 GHz ( lower panel). 

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In the text 
Figure 14: Upper panel: the frequency dependence of the absolute value of the ratio of the second order correction to the thermal SZE at the center of A2199 to the nonthermal SZE of an electron population that has a double powerlaw spectrum with parameters s_{1}=0.1, s_{2}=3.0, p_{1}=1, and and pressure . Lower panel: the frequency dependence of the ratio of the second order correction to the first order description of the thermal SZE. 

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In the text 
Figure 15: Upper panel: the frequency dependence of the absolute value of the ratio of the kinematic SZE for a value of the peculiar velocity of km s^{1} to the nonthermal SZE for an electron population with the same spectrum as in Fig. 14. Lower panel: the frequency dependence of the absolute value of the ratio of the kinematic SZE to the thermal SZE evaluated to first order. 

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In the text 
Figure 16: The thermal SZE spectra of the Perseus cluster evaluated at different projected radii of 0, 20, 50, 75, 100, 150, and 200 kpc ( from top to bottom) are shown in the frequency range 200500 GHz. For each curve, the SZE sampling in the frequency range 300450 GHz has an uncertainty of 1%. 

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In the text 
Figure 17: The radial profile of the projected temperature, derived from the fit to the SZE spectra in Perseus in the frequency range 300440 GHz with uncertainties of 1% (see Fig. 16), is shown as a function of the projected radius (in arcmin). 

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In the text 
Figure 18: The deprojected temperature radial profile in Perseus derived from the fit to the Xray data by Chrurazov et al. (2003) (solid curve) is compared to the fit to the SZE simulated data (dashed curve) shown in Fig. 17. For the SZE temperature profile, we use the radial profile given by Eq. (28), with keV, keV, kpc, and . 

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In the text 
Figure 19: As Fig. 17 but with uncertainties assumed at the level of 0.1% of the sampled SZE data points (see Fig. 16). 

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In the text 
Figure 20: The thermal SZE spectrum for the Ophiuchus cluster is shown at various projected radii of 0, 35, 70, 120, 200, 300 and 500 kpc ( from top to bottom). The assumed uncertainty in the SZE data points is 1%. 

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In the text 
Figure 21: The radial profile of the projected temperature ( upper panel) and of the IC gas density ( lower panel) as derived from the fit to the SZE spectra for Ophiuchus shown as a function of the projected radius (in arcmin). We use SZE data points in the frequency range 300400 GHz with uncertainties of 1% (see Fig. 20). 

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In the text 
Figure 22: The IC gas density profile of Ophiuchus derived from Xray observations (solid curve) compared with that derived from the optical depth radial profile shown in Fig. 21 (dashed line) obtained from SZE simulated observations (see text for details). 

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In the text 
Figure A.1: The SZE spectrum calculated following the method of W79 (solid line) and the method of BL09 (dashed line). The horizontal axis shows the adimensional frequency , while the vertical axis shows the SZE brightness change in units of . No difference between the two approaches is apparent. 

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In the text 
Figure A.2: Same as Fig. A.1 but with a zoom in the microwave frequency range at which the SZE telescope operate. In this lowfrequency range, there is also no difference between the two approaches. 

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In the text 
Figure A.3: The comparison of the ICS emission evaluated following the standard computational approach with the inclusion of the KleinNishina crosssection (solid line) and that evaluated following the method of BL09 in the Thomson approximation (dashed line). No difference appears between the two approaches up to a frequency . 

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In the text 
Copyright ESO 2010
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