X-ray and SZ scaling relations

X-ray and Sunyaev-Zel’dovich scaling relations in galaxy clusters

Abstract

We present an analysis of the scaling relations between X-ray properties and Sunyaev-Zel’dovich (SZ) parameters for a sample of 24 X-ray luminous galaxy clusters observed with Chandra and with measured SZ effect. These objects are in the redshift range 0.14–0.82 and have X-ray bolometric luminosity erg s, with at least 4000 net counts collected for each source. We perform a spatially resolved spectral analysis and recover the density, temperature and pressure profiles of the intra-cluster medium (ICM), just relying on the spherical symmetry of the cluster and the hydrostatic equilibrium hypothesis. The combined analysis of the SZ and X-ray scaling relations is a powerful tool to investigate the physical properties of the clusters and their evolution in redshift, by tracing out their thermodynamical history. We observe that the correlations among X-ray quantities only are in agreement with previous results obtained for samples of high- X-ray luminous galaxy clusters. On the relations involving SZ quantities, we obtain that they correlate with the gas temperature with a logarithmic slope significantly larger than the predicted value from the self-similar model. The measured scatter indicates, however, that the central Compton parameter is a proxy of the gas temperature at the same level of other X-ray quantities like luminosity. Our results on the X-ray and SZ scaling relations show a tension between the quantities more related to the global energy of the system (e.g. gas temperature, gravitating mass) and the indicators of the structure of the ICM (e.g. gas density profile, central Compton parameter ). Indeed, by using a robust fitting technique, the most significant deviations from the values of the slope predicted from the self-similar model are measured in the , , , relations. When the slope is fixed to the self-similar value, these relations consistently show a negative evolution suggesting a scenario in which the ICM at higher redshift has lower both X-ray luminosity and pressure in the central regions than the expectations from self-similar model. These effects are more evident in relaxed clusters in the redshift range 0.14-0.45, where a more defined core is present and the assumed hypotheses on the state of the ICM are more reliable.

keywords:
galaxies: clusters: general – cosmic microwave background – cosmology: observations – X-ray: galaxies: clusters
12

1 Introduction

Clusters of galaxies represent the largest virialized structures in the present universe, formed at relatively late times. The hierarchical scenario provides a picture in which the primordial density fluctuations generate proto-structures which are then subject to gravitational collapse and mass accretion, producing larger and larger systems. The cosmic baryons fall into the gravitational potential of the cluster dark matter (DM) halo formed in this way, while the collapse and the subsequent shocks heat the intra-cluster medium (ICM) up to the virial temperature ( keV).

In the simplest scenario which neglects all non-radiative processes, the gravity, which has not preferred scales, is the only responsible for the physical properties of galaxy clusters: for this reason they are expected to maintain similar properties when rescaled with respect to their mass and formation epoch. This allows to build a very simple model to relate the physical parameters of clusters: the so-called self-similar model (Kaiser, 1986; Evrard & Henry, 1991). Based on that, we can derive scaling relations (see Sect. 3) between X-ray quantities (like temperature , mass , entropy and luminosity ), and between X-ray and Sunyaev-Zel’dovich (SZ) measurements (like the Compton- parameter), thanks to the assumption of spherical collapse for the DM halo and hydrostatic equilibrium of the gas within the DM gravitational potential. These relations provide a powerful test for the adiabatic scenario. In particular, in the recent years the studies about the X-ray scaling laws (see, e.g., Allen & Fabian, 1998; Markevitch, 1998; Ettori et al., 2004b; Arnaud et al., 2005; Vikhlinin et al., 2005; Kotov & Vikhlinin, 2005), together with observations of the entropy distribution in galaxy clusters (see, e.g., Ponman et al., 1999; Ponman et al., 2003) and the analysis of simulated systems including cooling and extra non-gravitational energy injection (see, e.g., Borgani et al., 2004) have suggested that the simple adiabatic scenario is not giving an appropriate description of galaxy clusters. In particular the most significant deviations with respect to the self-similar predictions are: (i) a lower (by per cent) normalization of the relation in real clusters with respect to adiabatic simulations (Evrard et al., 1996); (ii) steeper slopes for the and relations; (iii) an entropy ramp in the central regions of clusters (see, e.g., Ponman et al., 1999; Ponman et al., 2003). These deviations are likely the evidence of non-radiative processes, like non-gravitational heating due to energy injection from supernovae, AGN, star formation or galactic winds (see, e.g., Pearce et al., 2001; Tozzi & Norman, 2001; Bialek et al., 2001; Babul et al., 2002; Borgani et al., 2002; Brighenti & Mathews, 2006) or cooling (see, e.g., Bryan, 2000). More recently some authors pointed out that there is a mild dependence of the X-ray scaling relations on the redshift, suggesting that there should be an evolution of these non-gravitational processes with (Ettori et al., 2004b).

An additional and independent method to evaluate the role of radiative processes is the study of the scaling relations based on the thermal SZ effect (Sunyaev & Zeldovich, 1970), which offers a powerful tool for investigating the same physical properties of the ICM, being the electron component of cosmic baryons responsible of both the X-ray emission and the SZ effect. The advantage of the latter on the former is the possibility of exploring clusters at higher redshift, because of the absence of the cosmological dimming. Moreover, since the SZ intensity depends linearly on the density, unlike the X-ray flux, which depends on the squared density, with the SZ effect it is possible to obtain estimates of the physical quantities of the sources reducing the systematic errors originated by the presence of sub-clumps and gas in multi-phase state and to study in a complementary way to the X-ray analysis the effects of extra-physics on the collapse of baryons in cluster dark matter halos, both via numerical simulations (White et al., 2002; da Silva et al., 2004; Diaferio et al., 2005; Nagai, 2006) and observationally (Cooray 1999; McCarthy et al. 2003a,b; Benson et al. 2004; LaRoque et al. 2006; Bonamente et al. 2006).

The main purpose of this paper is to understand how these SZ and X-ray scaling relations evolve with redshift. In particular we want to quantify how much they differ from the self-similar expectations in order to evaluate the amplitude of the effects of the non-gravitational processes on the physical properties of ICM. Another issue we want to debate is which relations can be considered a robust tool to link different cluster physical quantities: this has important consequences on the possibility of using clusters as probes for precision cosmology. To do that, we have assembled a sample of 24 galaxies clusters, for which measurements of the Compton- parameter are present in the literature. Respect the previous works we have done our own spatially resolved X-ray analysis recovering X-ray and SZ quantity necessary to investigate scaling relations. We have performed a combined spatial and spectral analysis of the X-ray data, which allows us to derive the radial profile for temperature, pressure, and density in a robust way. These results, which have high spatial resolution, rely only on the hydrostatic equilibrium hypothesis and spherical geometry of the sources. Moreover we can compare the observed physical quantities with the results of hydrodynamical numerical simulations in a consistent way.

The paper is organized as follows. In Sect. 2 we introduce our cluster sample and we describe the method applied to determine the X-ray properties (including the data reduction procedure) and the corresponding SZ quantities. In Sect. 3 we report our results about the scaling relations here considered, including the presentation of the adopted fitting procedure. Sect. 4 is devoted to a general discussion of our results, while in Sect. 5 we summarize our main conclusions. We leave to the appendices the discussion of some tecnical details of our data reduction procedure.

Hereafter we have assumed a flat cosmology, with matter density parameter , cosmological constant density parameter , and Hubble constant . Unless otherwise stated, we estimated the errors at the 68.3 per cent confidence level.

2 The dataset

2.1 Data reduction

We have considered a sample of galaxy clusters for which we have SZ data from the literature and X-ray data from archives (see Tables 1 and 2, respectively). In particular, we have considered the original sample of McCarthy et al. (2003b), to which we added two more objects from the sample discussed by Benson et al. (2004). For all these clusters we have analyzed the X-ray data extracted from the Chandra archive. In total we have 24 galaxy clusters with redshift ranging between 0.14 and 0.82, emission-weighted temperature in the range 6-12 keV and X-ray luminosity between and erg s. In the whole sample we have 11 cooling core clusters and 13 no-cooling core ones (hereafter CC and NCC clusters, respectively) defined according to the criterion that their cooling time in the inner regions is lower than the Hubble time at the cluster redshift.

 name
(mJy) (mJy)
A1413 0.99
A2204 0.79
A1914 1.20
A2218 1.03
A665 0.92
A1689 0.94
A520 1.10
A2163 0.74
A773 0.95
A2261 0.92
A2390 0.75
A1835 0.80
A697 0.96
A611 1.02
Zw3146 0.92
A1995 1.06
MS1358.4+6245 0.75
A370 1.19
RXJ2228+2037 0.88
RXJ1347.5-1145 0.70
MS0015.9+1609 0.97
MS0451.6-0305 1.31
MS1137.5+6625 1.16
EMSS1054.5-0321 1.04
Table 1: The SZ parameters for the galaxy clusters in our sample. For each object different columns report the name, the central value () of the Compton -parameter, the SZ flux integrated up to an overdensity of and over a fixed solid angle arcmin ( and , respectively) divided by the function (see eq. 9), and the parameter (see text). For two objects (namely A1914 and RXJ2228+2037) the corresponding errors are not provided by McCarthy et al. (2003b): in the following analysis we will assume for them a formal 1 error of 20 per cent.

We summarize here the most relevant aspects of the X-ray data reduction procedure. Most of the observations have been carried out using ACIS–I, while for 4 clusters (A1835, A370, MS0451.6-0305, MS1137.5+6625) we have data from the Back Illuminated S3 chip of ACIS–S. We have reprocessed the event 1 file retrieved from the Chandra archive with the CIAO software (version 3.2.2) distributed by the Chandra X-ray Observatory Centre. We have run the tool aciss_proces_ events to apply corrections for charge transfer inefficiency (for the data at 153 K), re-computation of the events grade and flag background events associated with collisions on the detector of cosmic rays. We have considered the gain file provided within CALDB (version 3.0) in this tool for the data in FAINT and VFAINT modes. Then we have filtered the data to include the standard events grades 0, 2, 3, 4 and 6 only, and therefore we have filtered for the Good Time Intervals (GTIs) supplied, which are contained in the flt1.fits file. We checked for unusual background rates through the script analyze_ltcrv, so we removed those points falling outside from the mean value. Finally, we have applied a filter to the energy (300-9500 keV) and CCDs, so as to obtain an events 2 file.

2.2 Spatial and spectral analysis

The images have been extracted from the events 2 files in the energy range (0.5-5.0 keV), corrected by using the exposure map to remove the vignetting effects, by masking out the point sources. So as to determine the centroid () of the surface brightness we have fitted the images with a circular one-dimensional (1D) isothermal -model (Cavaliere & Fusco-Femiano, 1976), by adding a constant brightness model, and leaving and free as parameters in the best fit. We constructed a set of () circular annuli around the centroid of the surface brightness up to a maximum distance (also reported in Table 2), selecting the radii according to the following criteria: the number of net counts of photons from the source in the (0.5-5.0 keV) band is at least 200-1000 per annulus and the signal-to-noise ratio is always larger than 2. The background counts have been estimated from regions of the same exposure which are free from source emissions.

Figure 1: The radial profiles for the projected temperature , normalized using the cooling-core corrected temperature , and for density are shown for all objects of our sample in the left and right panels, respectively. Solid and dashed lines refer to clusters with or without a central cooling flow, respectively

The spectral analysis has been performed by extracting the source spectra from () circular annuli of radius around the centroid of the surface brightness. We have selected the radius of each annulus out to a maximum distance (reported in Table 2), according to the following criteria: the number of net counts of photons from the source in the band used for the spectral analysis is at least 2000 per annulus and corresponds to a fraction of the total counts always larger than 30 per cent.

name obs. ACIS scale CC/
mode (ks) (kpc) (kpc) (kpc) (keV) () NCC (keV) () (erg/s)
A1413 0.143 1661 I 9.7 2.2 151 1111 1359 CC
A2204 0.152 6104 I 9.6 5.7 159 1183 1262 CC
A1914 0.171 3593 I 18.8 0.9 175 1449 1576 NCC
A2218 0.176 1666 I 36.1 3.2 179 1231 1320 NCC
A665 0.182 3586 I 29.1 4.2 184 1589 1476 NCC
A1689 0.183 1663 I 10.6 1.8 185 1446 1059 CC
A520 0.199 4215 I 66.2 3.5 197 1327 1455 NCC
A2163 0.203 1653 I 71.1 17.5 200 1846 1807 NCC
A773 0.217 5006 I 19.8 1.4 211 1105 1384 NCC
A2261 0.224 5007 I 24.3 3.3 216 1588 1595 CC
A2390 0.232 4193 S 92.0 8.3 222 1205 873 CC
A1835 0.253 495 S 10.3 2.3 237 914 970 CC
A697 0.282 4217 I 19.5 1.0 256 1865 1679 NCC
A611 0.288 3194 S 35.1 5.0 260 969 1172 CC
Zw3146 0.291 909 I 46.0 3.0 262 1061 1287 CC
A1995 0.319 906 S 44.5 1.4 279 877 914 CC
MS1358.4+6245 0.327 516 S 34.1 3.2 283 796 813 CC
A370 0.375 515 S 48.6 3.1 310 926 762 NCC
RXJ2228+2037 0.421 3285 I 19.8 4.9 332 1320 1636 NCC
RXJ1347.5-1145 0.451 3592 I 57.7 4.9 346 1558 1560 CC
MS0015.9+1609 0.546 520 I 67.4 4.1 383 1889 849 NCC
MS0451.6-0305 0.550 902 S 41.1 5.1 385 1092 1325 NCC
MS1137.5+6625 0.784 536 I 116.4 3.5 447 706 880 NCC
EMSS1054.5-0321 0.823 512 S 71.1 3.6 455 763 895 NCC
Table 2: The X-ray properties of the galaxy clusters in our sample. For each object different columns report the name, the redshift , the identification number of the Chandra observation, the used ACIS mode, the exposure time , the neutral hydrogen absorption (the labels and refer to objects for which has been fixed to the Galactic value or thawed, respectively), the physical scale corresponding to 1 arcmin, the maximum radii used for the spatial and for the spectral analysis ( and , respectively), the emission-weighted temperature , the metallicity (in solar units), a flag for the presence or not of a cooling core (labeled CC and NCC, respectively), the mass-weighted temperature , the gas mass , and the bolometric X-ray luminosity . The last three columns refer to an overdensity of . Sources extracted from the McCarthy et al. (2003b) sample and from the Benson et al. (2004) sample are indicated by apices (1) and (2), respectively.

The background spectra have been extracted from regions of the same exposure in the case of the ACIS–I data, for which we always have some areas free from source emission. Conversely, for the ACIS–S data we have considered the ACIS-S3 chip only and we have equally used the local background, but we have checked for systematic errors due to possible source contamination of the background regions. This is done considering also the ACIS “blank-sky” background files, which we have re-processed if their gain file does not match the one of the events 2 file; then we have applied the aspect solution files of the observation to the background dataset by using reproject_events, so as to estimate the background for our data. We have verified that the spectra produced by the two methods are in good agreement, and at last we decided to show only the results obtained using the local background.

All the point sources has been masked out by visual inspection. Then we have calculated the redistribution matrix files (RMF) and the ancillary response files (ARF) for each annulus: in particular we have used the tools mkacisrmf and mkrmf (for the data at 120 K and at 110 K, respectively) to calculate the RMF, and the tool mkarf to derive the ARF of the regions.

For each of the annuli the spectra have been analyzed by using the package XSPEC (Arnaud, 1996) after grouping the photons into bins of 20 counts per energy channel (using the task grppha from the FTOOLS software package) and applying the -statistics. The spectra are fitted with a single-temperature absorbed MEKAL model (Kaastra, 1992; Liedahl et al., 1995) multiplied by a positive absorption edge as described in Vikhlinin et al. (2005): this procedure takes into account a correction to the effective area consisting in a 10 per cent decrement above 2.07 keV. The fit is performed in the energy range 0.6-7 keV (0.6-5 keV for the outermost annulus only) by fixing the redshift to the value obtained from optical spectroscopy and the absorbing equivalent hydrogen column density to the value of the Galactic neutral hydrogen absorption derived from radio data (Dickey & Lockman, 1990), except for A520, A697, A2163, MS1137.5+6625, MS1358.4+6245 and A2390, where we have decided to leave free due to the inconsistency between the tabulated radio data and the spectral fit result. Apart for these objects where also the Galactic absorption is left free, we consider three free parameters in the spectral analysis for th annulus: the normalization of the thermal spectrum , the emission-weighted temperature ; the metallicity retrieved by employing the solar abundance ratios from Anders & Grevesse (1989). The best-fit spectral parameters are listed in Table 2.

The total (cooling-core corrected) temperature has been extracted in a circular region of radius , with , centred on the symmetrical centre of the brightness distribution. In the left panel of Fig. 1 we present for all clusters of our sample the projected temperature profile () normalized by as a function of the distance from the centre , given in units of , where is the radius corresponding to an ovedensity of .

2.3 Spectral deprojection analysis

To measure the pressure and gravitating mass profiles in our clusters, we deproject the projected physical properties obtained with the spectral analysis by using an updated and extended version of the technique presented in Ettori et al. (2002) and discussed in full detail in Appendix A. Here we summarize briefly the main characteristics of the adopted technique: (i) the electron density is recovered both by deprojecting the surface brightness profile and the spatially resolved spectral analysis obtaining a few tens of radial measurements; (ii) once a functional form of the DM density profile , where are free parameters of the DM analytical model, and the gas pressure at are assumed, the deprojected gas temperature, , is obtained by integration of the hydrostatic equilibrium equation:

(1)

where is the average molecular weight, is the proton mass. So expressed in keV units. In the present study, to parametrize the cluster mass distribution, we consider two models: the universal density profile proposed by Navarro et al. (1997) (hereafter NFW) and the one suggested by Rasia et al. (2004) (hereafter RTM).

The NFW profile is given by

(2)

where is the critical density of the universe at redshift , , , and

(3)

where is the concentration parameter, is the scale radius, , .

The RTM mass profile is given by:

(4)

with , where is a reference radius and is given by:

(5)

So we have and for the NFW and RTM models, respectively.

Figure 2: Example of temperature spectral deprojection for cluster A1413. We display the two quantities which enter in the eq. 43 in the spectral deprojection analysis to retrieve the physical parameters: the observed spectral projected temperature (stars with errorbars) and the theoretical projected temperature (triangules, indicated as in Appendix A). We also show the theoretical deprojected temperature (points), which generates through convenient projection tecniques.

The comparison of the observed projected temperature profile (Sect. 2.2) with the deprojected (eq. 43 in Appendix A), once the latter has been re-projected by correcting for the temperature gradient along the line of sight as suggested in Mazzotta et al. (2004), provides the best estimate of the free parameters through a minimization, and therefore of (see an example in Figure 2).

In the right panel of Fig. 1 we present the density profiles (plotted versus ) as determined through the previous method. In general, we find there is no significant effect on the determination of the physical parameters when adopting the two different DM models. Hereafter we will use the physical parameters determined using the RTM model, reported with their corresponding errors in Table 2, where we also list the exposure time, the number and the instrument (ACIS–I or ACIS–S) used for each of the Chandra observations.

Finally we computed the total mass enclosed in a sphere of radius as where the radius corresponds to a given overdensity : we considered the cases where the overdensity is equal to and . The values for masses and radii, together with the parameters for the RTM model, are reported in Table 3. The errors on the different quantities represent the 68.3 per cent confidence level and are computed by looking to the regions in the parameter space where the reduction of with respect to its minimum value is smaller than a given threshold, fixed according to the number of degrees of freedom d.o.f. (see, e.g., Press et al., 1992). Notice that we included in the eq.(1) the statistical errors related to measurement errors of .

name (d.o.f.)
(kpc) ( erg cm) () (kpc) () (kpc)
A1413 5.29(3) 2.30 520 5.58 1195
A2204 4.24(5) 6.74 742 19.12 1796
A1914 2.29(5) 3.39 586 5.99 1212
A2218 0.61(2) 2.14 502 4.36 1088
A665 1.23(5) 1.88 480 7.77 1317
A1689 0.68(4) 4.14 624 9.40 1402
A520 0.08(3) 3.72 599 12.66 1540
A2163 3.00(5) 4.07 616 52.58 2472
A773 1.38(2) 2.01 485 4.54 1087
A2261 2.82(3) 2.68 532 6.17 1201
A2390 23.08(4) 6.59 716 33.22 2099
A1835 0.80(1) 4.09 606 10.95 1439
A697 1.17(4) 3.23 554 10.46 1402
A611 0.81(3) 2.11 480 5.18 1107
Zw3146 4.36(3) 5.41 656 22.50 1804
A1995 3.05(2) 3.51 562 14.96 1558
MS1358.4+6245 0.60(1) 3.62 566 17.37 1633
A370 4.21(1) 3.10 528 10.58 1359
RXJ2228+2037 0.12(2) 1.59 415 4.90 1033
RXJ1347.5-1145 3.58(5) 9.49 744 24.00 1734
MS0015.9+1609 0.96(4) 1.72 406 9.75 1237
MS0451.6-0305 0.14(5) 3.68 522 11.89 1320
MS1137.5+6625 2.12(1) 1.91 382 5.47 928
EMSS1054.5-0321 0.03(1) 2.17 393 27.21 1560
Table 3: Different physical properties for the clusters in our sample. For each object the different columns report the name, the minimum value for (with the corresponding number of degrees of freedom d.o.f.), the virial radius , the reference scale , the value of the pressure , the mass and the radius corresponding to an overdensity of ( and , respectively), the mass and the radius corresponding to an overdensity of ( and , respectively). All quantities are derived by assuming the RTM model.

2.4 Determination of the X-ray properties

The bolometric X-ray luminosity has been calculated by correcting the observed luminosity determined from the spectral analysis performed by XSPEC excluding the central cooling region of 100 kpc (the results are reported in Table 2):

(6)

where , , and are the best-fit parameters of the -model on the image brightness, is the normalization of the thermal spectrum drawn with XSPEC, and corrected for the emission from the spherical source up to 10 Mpc intercepted by the line of sight: , with , , , and .

The gas mass enclosed in a circular region having overdensity has been computed from the total gas density , that we directly obtained from the spectral deprojection, up to . We have checked that the exclusion of the central cooling region does not significantly affect the resulting values for .

Finally we have estimated the total mass-weighted temperature:

(7)

which can be compared to the total emission-weighted temperature ; represents the number of annuli inside . Notice that our average deprojected temperature profile implies the following relation between the maximum, the deprojected and the mass-weighted temperatures: ( for the CC-only subsample). The physical parameters obtained in this way are also listed in Table 2 for all clusters of our sample.

2.5 Determination of the Sunyaev-Zel’dovich properties

The thermal SZ (Sunyaev & Zeldovich, 1970) effect is a very small distortion of the spectrum of the cosmic microwave background (CMB), due to the inverse Compton scatter between cold CMB photons and hot ICM electrons (for recent reviews see, e.g., Birkinshaw, 1999; Carlstrom et al., 2002; Rephaeli et al., 2005). This comptonization process statistically rises the photon energy, producing a distortion of the CMB black-body spectrum. The final result is a decrease (increase) of the CMB flux at frequencies smaller (larger) than about 218 GHz. The amplitude of this effect is directly proportional to the Compton parameter , which is defined as

(8)

where is the angular distance from the cluster centre, is the Thomson cross-section, and is the pressure of the ICM electrons at the position r; the integral is done along the line of sight.

The SZ effect can be expressed as a change in the brightness:

(9)

respectively; here , , is the present CMB temperature and the function is given by:

(10)

and accounts for the frequency dependence of the SZ effect; the term represents the relativistic correction (see, e.g., Itoh et al., 1998), which, however, is negligible for clusters having keV.

We consider the Compton- parameter integrated over the entire solid angle (and given in flux units) defined as:

(11)

To remove the dependence of on the angular diameter distance we use the intrinsic integrated Compton parameter , defined as:

(12)

The same quantity, but integrated over a fixed solid angle , can be similarly written as:

(13)

We fixed arcmin, that is than the field of view of OVRO, used in the observations of most of the sources in our sample (see, e.g., McCarthy et al., 2003a). Notice that in order to remove the frequency dependence we have normalized , and to .

To integrate eqs. (11) and (13) we have recovered from eq. 8 by using the pressure profile determined in the spectral analysis (Sect. 2.3), renormalized in such a way that equals the central Comption parameter taken from the literature. This method can lead to systematics on and due to the fact that, even if we are assuming the true pressure profiles in eq. (8), has been obtained by assuming an isothermal -model inferred from the brightness profile. The value of is thus potentially dependent on the underlying model of . As discussed in recent works (see, e.g., LaRoque et al., 2006; Bonamente et al., 2006), the relaxation of the isothermal assumption should apply to the analysis of both X-ray and SZ data, to obtain a robust and consistent description of the physics acting inside galaxy clusters. Unfortunately, we have only the central Compton parameter, and not the complete data, which are not public available: so it is very difficult to quantify the amplitude of this systematics, being determined through a best fit in the plane.
Nevertheless, we can give an estimate in this way: we have computed the central Compton parameter inferred from the X-ray data by parametrizing first in eq. (8) with a -model inferred on the brightness images:

(14)

with derived from the brightness profile :

(15)

where is the X-ray cooling function of the ICM in the cluster rest frame in cgs units (erg cm s) integrated over the energy range of the brightness images ( keV). Then we have calculated by accounting in eq. 8 for the true pressure profile recovered by the spectral deprojection analysis (Sect. 2.3), and therefore we determined the ratio . We notice that the parameter differs from the unity of per cent, comparable to statistical errors.

The different quantities related to the SZ effect are listed in Table 1 for all clusters in our sample.

3 The X-ray and SZ scaling relations: theory and fitting procedure

3.1 The scaling relations in the self-similar model

The self-similar model (see, e.g., Kaiser, 1986) gives a simple picture of the process of cluster formation in which the ICM physics is driven by the infall of cosmic baryons into the gravitational potential of the cluster DM halo. The collapse and subsequent shocks heat the ICM up to the virial temperature. Thanks to this model, which assumes that gravity is the only responsible for the observed values of the different physical properties of galaxy clusters, we have a simple way to establish theoretical analytic relations between them.

Numerical simulations confirm that the DM component in clusters of galaxies, which represents the dominant fraction of the mass, has a remarkably self-similar behaviour; however the baryonic component does not show the same level of self-similarity. This picture is confirmed by X-ray observations, see for instance the deviation of the relation in clusters, which is steeper than the theoretical value predicted by the previous scenario. These deviations from self-similarity have been interpreted as the effects of non-gravitational heating due to radiative cooling as well as the energy injection from supernovae, AGN, star formation or galactic winds (see, e.g., Tozzi & Norman, 2001; Bialek et al., 2001; Borgani et al., 2002; Babul et al., 2002; Borgani et al., 2004; Brighenti & Mathews, 2006) which make the gas less centrally concentrated and with a shallower profile in the external regions with respect the DM component. Consequently, the comparison of the self-similar scaling relations to observations allows us to evaluate the importance of the effects of the non-gravitational processes on the ICM physics.

For and we have the following dependences on the cosmology:

(16)

and

(17)

respectively, where the factor , with , accounts for evolution of clusters in an adiabatic scenario (Bryan & Norman, 1998).

Assuming the spherical collapse model for the DM halo and the equation of hydrostatic equilibrium to describe the distribution of baryons into the DM potential well, in the self-similar model the cluster mass and temperature are related by:

(18)

so we have . By setting , from the previous equations we can easily obtain the following relations (see, e.g., Markevitch, 1998; Allen & Fabian, 1998; Ettori et al., 2004b; Arnaud et al., 2005; Diaferio et al., 2005; Vikhlinin et al., 2005; Kotov & Vikhlinin, 2005):

(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)

We also remember here that for galaxy clusters similar scaling laws exist also in the X-ray band (see, e.g., Ettori et al., 2004a; Arnaud et al., 2005; Kotov & Vikhlinin, 2005; Vikhlinin et al., 2006):

(27)
(28)
(29)
(30)
(31)

In our work we have considered all the physical quantity at fixed overdensity (), i.e. in the above equations.

3.2 Fitting the scaling relations

We describe here the method adopted to obtain the best-fitting parameters in the scaling relations. Since they are power-law relations, we carry out a log-log fit:

(32)

where and represent the independent and dependent variables, respectively (hereafter ); and are the two free parameters to be estimated. However, in the considered scaling relations it is unclear which variable should be considered as (in)dependent. Moreover both - and -data have errors due to measurement uncertainties, plus an intrinsic scatter. For these reasons, the ordinary least squares (OLS) minimization approach is not appropriate: in fact it does not take into account intrinsic scatter in the data, and it is biased when errors affect the independent variable. So we decided to use the BCES (Bivariate Correlated Errors and intrinsic Scatter) modification or the bisector modification BCES proposed by Akritas & Bershady (1996), for which the best-fit results correspond to the bisection of those obtained from minimizations in the vertical and horizontal directions. Both these methods are robust estimators that take into account both any intrinsic scatter and the presence of errors on both variables.

The results for the best-fit normalization and slope for the listed scaling relations are presented in Table 4, where we also report the values of the total scatter

(33)

and of the intrinsic scatter calculated as:

(34)

where , with being the statistical error of the measurement , and is the number of degrees of freedom (, with equal to total number of data).

Notice that in these fits the physical quantities (, , , ) refer to estimated through the mass estimates based on the RTM model.

Cooling core clusters All clusters
11 objects 24 objects
relation method
1.22()/1.67 -1.07() 0.090 0.113 1.19()/1.67 -0.91() 0.116 0.137 (1)
0.93()/1.00 0.61() 0.033 0.076 0.92()/1.00 0.66() 0.120 0.140 (1)
2.21()/1.50 0.19() 0.138 0.154 2.06()/1.50 0.15() 0.123 0.141 (2)
1.25()/1.00 -0.50() 0.248 0.257 1.22()/1.00 -0.41() 0.211 0.222 (2)
0.75()/0.75 -0.69() 0.156 0.170 0.61()/0.75 -0.48() 0.124 0.142 (2)