AMiBA SZE Properties and Scaling Relations

# AMiBA: Sunyaev-Zel’dovich Effect Derived Properties and Scaling Relations of Massive Galaxy Clusters

###### Abstract

The Sunyaev-Zel’dovich Effect (SZE) has been observed toward six massive galaxy clusters, at redshifts in the 86-102 GHz band with the Y. T. Lee Array for Microwave Background Anisotropy (AMiBA). We modify an iterative method, based on the isothermal -models, to derive the electron temperature , total mass , gas mass , and integrated Compton within , from the AMiBA SZE data. Non-isothermal universal temperature profile (UTP) models are also considered in this paper. These results are in good agreement with those deduced from other observations. We also investigate the embedded scaling relations, due to the assumptions that have been made in the method we adopted, between these purely SZE-deduced , , and . Our results suggest that cluster properties may be measurable with SZE observations alone. However, the assumptions built into the pure-SZE method bias the results of scaling relation estimations and need further study.

cosmology: observation — galaxies: clusters: — sunyaev-zeldovich effect:

## 1. Introduction

The Sunyaev-Zel’dovich Effect (SZE) is an useful tool for studies of galaxy clusters. This distortion of the Cosmic Microwave Background (CMB) is caused by the inverse Compton scattering by high energy electrons as the CMB propagates through the hot plasma of galaxy clusters (Sunyaev & Zel’dovich, 1972). The SZE signal is essentially redshift independent, making it particularly useful for determining the evolution of large-scale structure.

For upcoming SZE cluster surveys (Ruhl et al., 2004; Fowler, 2004; Kaneko, 2006; Ho et al., 2009), it is important to investigate the relations between SZE flux density and other cluster properties, such as mass, temperature, and gas fraction. By assuming that the evolution of clusters is dominated by self-similar gravitational processes, we can predict simple power law relations between integrated Compton and other cluster properties (Kaiser, 1986). Strong correlations between integrated SZE flux and the mass of clusters are also suggested by numerical simulations (da Silva et al., 2004; Motl et al., 2005; Nagai, 2006). These relations imply the possibility of determining the masses and temperatures of clusters, and investigating cluster evolution at high redshift, with SZE observation data alone.

Joy et al. (2001) and Bonamente et al. (2008) demonstrated an iterative approach based on the isothermal model to estimate the values of electron temperature , total mass , gas mass , and Compton- from SZE data alone. In this paper, we seek to derive the same cluster properties from the AMiBA SZE measurements of six clusters. Due to the limited space sampling, the AMiBA data do not provide useful constraints on the structural parameters, and , in a full iterative model fitting. Instead, we adopt and from published X-ray fits and use a Markov Chain Monte-Carlo (MCMC) method to determine the cluster properties ( and ). We also estimate these cluster properties from AMiBA data with structural constraints from X-ray data using the non-isothermal universal temperature profile model (Hallman et al., 2007). All quantities are integrated to spherical radius within which the mean over-density of the cluster is  times the critical density at the cluster’s redshift. We then investigate the scaling relations between these cluster properties derived from the SZE data, and identify correlations between those properties that are induced by the iterative method. We note that Huang et al. (2009) investigate the scaling relations between the values of Compton from AMiBA SZE data and other cluster properties from X-ray and other data. All results are in good agreement. However, we are concerned that there are embedded relations between the properties we derived using this method. Therefore, we also investigate the embedded scaling relations between SZE-derived properties as well.

We assume the large-scale structure of the Universe to be described by a flat CDM model with , , and Hubble constant , corresponding to the values obtained using the WMAP 5-year data (Dunkley et al., 2009). All uncertainties quoted are at the 68% confidence level.

## 2. Determination of cluster properties

### 2.1. AMiBA Observation of SZE

AMiBA is a coplanar interferometer (Ho et al., 2009; Chen et al., 2009). During 2007, it was operated with 7 close-packed antennas of 60 cm in diameter, giving 21 vector baselines in - space and a synthesized resolution of (Ho et al., 2009). The antennas are mounted on a six-meter platform (Koch et al., 2009), which we rotate during the observations to provide better - coverage. The observations of SZE clusters, the details about the transform of the data into calibrated visibilities, and the estimated cluster profiles are presented in Wu et al. (2009). Further system checks are discussed in Lin et al. (2009) and Nishioka et al. (2009). For other scientific results deduced from AMiBA 2007 observations, please refer to Huang et al. (2009); Liu et al. (2010); Koch et al. (2010); Molnar et al. (2010); Umetsu et al. (2009)

### 2.2. Isothermal β modeling

Because the - coverage is incomplete for a single SZE experiment, we can measure neither the accurate profile of a cluster nor its central surface brightness. Therefore we have chosen to assume an SZE cluster model and thus a surface brightness profile, so that a corresponding template in the - space can be fitted to the observed visibilities in order to estimate the underlying model parameters. We consider a spherical isothermal -model (Cavaliere & Fusco-Femiano, 1976, 1978), which expresses the electron number density profile as

 ne(r)=ne0(1+r2r2c)−3β/2, (1)

where is the central electron number density, is the radius from the cluster center, is the core radius, and is the power-law index.

Traditionally the SZE is characterized by the Compton parameter, which is defined as the integration along the line of sight with given direction,

 y(^n)≡∫∞0σTnekBTemec2dl. (2)

Compton is related to as

 ΔISZE=ICMByf(x,Te)xexex−1, (3)

where , is the present CMB specific intensity, and (e.g., LaRoque et al., 2006). is a relativistic correction (Challinor & Lasenby, 1998), which we take into account to first order in . The relativistic correction becomes significant when the electron temperature exceeds , which is the regime of our cluster sample.

One can combine Equations (1-3) and integrate along the line of sight to obtain the SZE in the apparent radiation intensity as

 ΔISZE=I0(1+θ2/θ2c)(1−3β)/2 (4)

where and are the angular equivalents of and respectively. Because the clusters in our sample are not well resolved by AMiBA, we cannot get a good estimate of , , and simultaneously from our data alone. Instead, we use the X-ray derived values for and , as summarized in Table 1, and then estimate the central specific intensity (Liu et al., 2010) by fitting Equation (4) to the calibrated visibilities obtained by Wu et al. (2009). In the analysis we take into account the contamination from point sources and structures in the primary CMB.

Given the -model described above, we can derive relations between cluster parameters and estimate them using the MCMC method. The parameters to be estimated are the electron temperature , , total mass , gas mass , and the integrated Compton .

Theoretically can be fomulated through the hydrostatic equilibrium equation (e.g., Grego et al., 2001; Bonamente et al., 2008):

 Mt(r2500)=3βkBTeGμmpr32500r2c+r22500, (5)

where is the gravitational constant and is the mean mass per particle of gas in units of the mass of proton, . To calculate , we assume that takes the value appropriate for clusters with solar metallicity as given by Anders & Grevesse (1989). Here we use the value . By combining Equation (5) and the definition of , we can obtain as a function of , , , and redshift (e.g., Bonamente et al., 2008)

 r2500= ⎷3βkBTeGμmp143πρc(z)⋅2500−r2c. (6)

Then can be expressed, by integrating the in Equation (1) as

 Mg(r)=4πμene0mpD3A∫r/DA0(1+θ2θ2c)−3β/2θ2dθ, (7)

where is the mean particle mass per electron in unit of , is the angular diameter determined by , and is the central electron density, derived through the equation in LaRoque et al. (2006):

 ne0=ΔT0mec2Γ(32β)f(x,Te)TCMBσTkBTeDAπ1/2Γ(32β−12)θc, (8)

where is the gamma function, is the SZE temperature change, and is the present CMB temperature. is derived as .

Finally, with the computed earlier and the estimated here we can integrate the Compton out to to yield

 Y=2πΔT0f(x,Te)TCMB∫θ25000(1+θ2θ2c)(1−3β)/2θdθ, (9)

where indicates the projected angular size of .

With the formulae as described above, for a set of , , and as measured from X-ray observations and from AMiBA SZE observation, we can arbitrarily assign a ‘pseudo’ electron temperature , and then determine the pseudo , , , and . Given and , we obtained the pseudo gas fraction . Using as a function of we applied the MCMC method by varying and to estimate the likelihood distribution of each cluster property. While estimating the MCMC likelihood we assume that the likelihoods of and are independent. The likelihood distributions of for each cluster are taken from the fitting results of Liu et al. (2010), while the likelihood distribution of is assumed to be Gaussian with mean and standard deviation , which is the ensemble average over 38 clusters observed by Chandra and OVRO/BIMA (LaRoque et al., 2006).

In the process, the values of , , and are taken from other observational results which are summarized in Koch et al. (2010) and Table 1. We took the model parameters from both ROSAT and Chandra X-ray results. The Chandra results were derived by fitting an isothermal model to the X-ray data with a central 100-kpc cut. The aim of the cut-off is to exclude the complicated non-gravitational physics (e.g, radiative cooling and feedback mechanisms) in cluster cores. Table  2 summarizes our results derived assuming an isothermal model. We present the results obtained with isothermal model parameters derived with and without 100-kpc cut both here. Figure 1 compares our results with the SZE-X-ray joint results obtained from OVRO/BIMA and Chandra data (Bonamente et al., 2008; Morandi et al., 2007). These are in good agreement.

### 2.3. Utp β model

The simulation done by Hallman et al. (2007) suggested incompatibility between isothermal model parameters fitted to X-ray surface brightness profiles and those fitted to SZE profiles. This incompatibility also causes bias in the estimates of and . They suggested a non-isothermal model with a universal temperature profile (UTP). We also considered how the UTP model changes our estimates of cluster properties in this section.

In the UTP -model, the baryon density profile is the same as Equation  (1), and the temperature profile can be written as (Hallman et al., 2007):

 Te(r)=⟨T⟩500T0(1+(rαr500)2)−δ, (10)

where indicates the average spectral temperature inside . , , and are dimensionless parameters in the universal temperature profile model. is the outer slope of the temperature profile, outside of a core with electron temperature . This core is of size . The total mass can be obtained by solving the hydrostatic equilibrium equation (Fabricant et al., 1980):

 Mt(r)=−kBr2Gμmp(Te(r)dne(r)dr+ne(r)dTe(r)dr). (11)

In the isothermal -model, Equation (11) can be reduced into the form of Equation (5). However, in the UTP -model, the derivative of with respect to in Equation (11) is no longer zero. By applying Equation (1) and Equation (10) in Equation (11), one can obtain:

 Mt(r)=kBTe0Gμmp(3βr3r2+r2c+2δr3r2+α2r2500)(1+r2α2r2500)−δ. (12)

By combining Equation (12) and the definition of , an analytical solution for can be obtained as:

 r500= ⎷(1+α2)(3βA−r2c)+2δA+√D2(1+α2), (13)

where , and . If or , which indicate the nearly isothermal case, Equation (13) reduces to a form similar to Equation (6).

Using the definition of , can be written as:

 Mt(r500)=500⋅43πr3500ρc(z). (14)

For an arbitrary overdensity , we can not find an analytical solution for arbitrary (i.e.: , , etc.). However, with the known , we can still find the numerical solution for easily. We can then solve for using Equation (12).

To yield the central electron number density, we consider the formula for the Compton resulting from the UTP -model (see the Appendix of Hallman et al. (2007)). By setting the projected radius in Equation (A10) in Hallman et al. (2007), one can obtain:

 (15)

where

 ISZ(0)=π1/2Γ(32β+δ−12)F2,1(δ,12;3β2+δ,1−r2cα2r2500)rcΓ(3β2+δ), (16)

and is Gauss’ hypergeometric function. Here we assume , and the change of due to the change of along line of sight is negligible. Actually, by numerical calculation we found that the error in Equation (15) caused by this assumption is less than . Because the UTP model assumes the electron density profile as same as the isothermal model, we can rewrite in UTP model by simply applying Equation (15) in Equation (7).

Thus, the integration of the Compton profile, instead of Equation (9), becomes:

 (17)

where and .

We were not able to constrain the parameters , , , and of the UTP significantly with our SZE data alone. However, the simulation of Hallman et al. (2007) suggested that there is no significant systematic difference between the values of and resulting from fitting an isothermal model to mock X-ray observations and those parameters fitted using the UTP model. Therefore, we simply assume that the ratio between the isothermal value and UTP value is , and , for each cluster. We also assume , , and . Those values are taken from the average of results of Hallman et al. (2007). Then we fit to AMiBA SZE observation data with the UTP model parameters above by fixing , , and , and treating the likelihood distributions of and as two independent Gaussian-distributions. Finally, we applied the MCMC method, which varies , , , and , to estimate cluster properties with the equations derived from the UTP model and the data fitting results.

Table  3 summarizes our results derived with the UTP model. Figure 2 compares our results with the SZE-X-ray joint results obtained from OVRO/BIMA and Chandra data (Bonamente et al., 2008; Morandi et al., 2007). These are also in good agreement. We find that the electron temperature derived with the UTP model are in significantly better agreement with the temperatures from Chandra X-ray measurements.

## 3. Embedded scaling relations

The self-similar model (Kaiser, 1986) predicts simple power-law scaling relations between cluster properties (e.g., Bonamente et al., 2008; Morandi et al., 2007). Motivated by this, people usually investigate the scaling relations between the derived cluster properties from observational data to see whether they are consistent with the self-similar model. However, the method described above is based on the isothermal -model and the UTP -model. Therefore, there could be some embedded relations which agree with self-similar model predictions between the derived properties. We investigated the embedded relations through both analytical and numerical methods.

### 3.1. Analytical formalism and numerical analysis

In the isothermal model, by applying Equation (6) in Equation (5), can be rewritten as

 Mt=2500⋅43πρc(z)⎛⎝3βkBTeGμmp12500⋅43πρc(z)−r2c⎞⎠32. (18)

As we can see, while is set to be a constant, and , which implies , the relation will be obtained. However, for some of the clusters we considered in this paper, the values of are only slightly above . Therefore, we have to investigate the scaling relation between and by considering .

By partially differentiating Equation (18) by , and multiplying it by , we can get

 ∂lnMt∂lnTe=32(r22500+r2c)r22500, (19)

which decreases from at to as . That implies behaves as while and while approaches infinity. This result shows that there is an embedded - relation consistent with the self-similar model in the method described above.

If we assume that the gas fraction is a constant, the scaling relation between and will be as same as the relation between and .

In order to investigate the relations between integrated and the other cluster properties, we consider Equation (9). By combining Equation (6)-(8), one can obtain:

 ΔT0=Mg(r2500)f(x,Te)TCMBσTkBTeΓ(32β−12)θc4π1/2μempD2Amec2Γ(32β)∫θ25000(1+θ2θ2c)−3β/2θ2dθ. (20)

Then we combine Equation (20) and Equation (9) and obtain:

 Y=π1/2Mg(r2500)σTkBTe2μempmec2D2Ag(θ2500,θc,β), (21)

where

 g(θ2500,θc,β)=Γ(32β−12)θc∫θ25000(1+θ2θ2c)(1−3β)/2θdθΓ(32β)∫θ25000(1+θ2θ2c)−3β/2θ2dθ (22)

is a dimensionless function of , , and .

We also calculated to investigate the behavior of when varies (see Figure 3). As we can see in Figure 3, varies between and while and . We also noticed that approaches as approaches infinity. This result indicates that behaviour similar to the self-similar model is built into scaling relation studies based solely on SZE data.

The effect of varying is investigated. If we consider power law scaling relation

 Q=10AXB (23)

between and with written as Equation (18), one can find that changing the value of will only affect the normalization factor . In other words, if we change to , will be changed to .

In the - relation, will affect the scaling power as shown in Figure 3. varies within a range of only while .

Considering the UTP model, we undertook a similar analysis of the embedded scaling relation. The results, which are similar with those obtained with the isothermal model, are shown in Figure 4.

### 3.2. Calculation of Scaling Relations

Here we investigate the -, -, and - scaling relations for the quantities derived above. We also study the - scaling relation with the from AMiBA SZE data and the from X-ray data (Bonamente et al., 2008; Morandi et al., 2007).

For a pair of cluster properties -, we consider the power-law scaling relation (Equation (23)). To estimate and , we perform a maximum-likelihood analysis in the log-log plane. For the - relation, because and are independent measurements from different observational data, we can simply perform linear minimum- analysis to estimate and (Press et al., 1992; Benson et al., 2004). On the other hand, for the SZE-derived properties, because they are correlated and so are their likelihoods (i.e., , as manifested by the colored areas in Figure 5), we cannot apply analysis. Instead we use a Monte Carlo method by randomly choosing one MCMC iteration from each cluster many times. With each set of iterations we derived a pair of and using linear regression method. Finally we estimate the likelihood distribution of and using the distribution of and . The results are presented in Table 4 and Figures 5 and 6. However, as we discussed in Section 3.1, the scaling relations between SZE-derived properties should be interpreted as a test of embedded scaling relations rather than estimations of the true scaling relations. On the other hand, the - relation compared and from different experiments. Therefore, we can regard it as a test of the scaling relation prediction.

## 4. Discussions and Conclusion

We derived the cluster properties, including , ,, and , for six massive galaxy clusters () mainly based on the AMiBA SZE data. These results are in good agreement with those obtained solely from the OVRO/BIMA SZE data, and those from the joint SZE-X-ray analysis of Chandra-OVRO/BIMA data. In the comparison, the SZE-X-ray joint analysis gives smaller error bars than the pure SZE results, because currently the uncertainty in the measurement of the SZE flux is still large. On the other hand, in our current SZE-based analysis, due to the insufficient - coverage of the 7-element AMiBA we still need to use X-ray parameters for the cluster model i.e., the and for the -model. However, Nord et al. (2009) have deduced and from an APEX SZE observation alone recently. For AMiBA, the situation will be improved when it expands to its 13-element configuration with 1.2m antennas (AMiBA13; Ho et al., 2009), and thus much stronger constraints on the cluster properties than current AMiBA results are expected. Furthermore, with about three times higher angular solution, we should be able to estimate and from our SZE data with AMiBA13 and make our analysis purely SZE based (Ho et al., 2009; Molnar et al., 2010). Nevertheless, the techniques of using SZE data solely to estimate cluster properties are still important, because many upcoming SZE surveys will observe SZE clusters for which no X-ray data are available (Ruhl et al., 2004; Fowler, 2004; Kaneko, 2006; Ho et al., 2009), especially for those at high redshifts.

Hallman et al. (2007) suggested that adopting the UTP model for SZE data on galaxy clusters will reduce the overestimation of the integrated Compton and gas mass. However, the values we obtained with the UTP model are not smaller than those obtained with the isothermal model. The values deduced using the UTP model are even larger than those deduced using the isothermal model.

For the case of integrated Compton Y, when we compare deduced using the UTP model , and those deduced using the isothermal model , we found that the are smaller than , as predicted by Hallman et al. (2007). The reason is that the Compton profile predicted using the UTP model will decrease more quickly than the profile predicted by the isothermal model, with increasing radius. Therefore, the ratio will decrease as decreases.

We also noticed that the electron temperature values obtained with the isothermal model are significantly higher than the temperatures deduced from X-ray data for most clusters we considered. The temperatures of clusters obtained using the UTP model are lower than those obtained with the isothermal model and thus are in better agreement with those deduced from X-ray data. Therefore, in the UTP model, with similar and lower temperature, we should get larger .

The electron temperatures derived using the UTP model are in better agreement with X-ray observation results than those derived using the isothermal model. This result implies that the UTP model may provide better estimates of the electron temperature when we can use only the model parameters from X-ray observation. However, we noticed that the UTP model produced larger errorbars than the isothermal model did. These increased errors are based on the uncertainties of and which we insert by hand. On the other hand, because we treat and as independent parameters in this work, the uncertainty could be over estimated due to the degeneracy between these two parameters. If we can access to the likelihood distributions of and of the UTP model derived from observation, the error-bars might be reduced significantly.

There is a concern that the scaling relations among the purely SZE-derived cluster properties may be implicitly embedded in the formalism we used here. In this paper, we also investigate for the first time the embedded scaling relations between the SZE-derived cluster properties. Our analytical and numerical analyses both suggest that there are embedded scaling relations between SZE-derived cluster properties, with both the isothermal model and the UTP model, while we fix . The embedded - and - scaling relations are close to the predictions of self-similar model. The results imply that the assumptions built in the pure-SZE method significantly affect the scaling relation between the SZE-derived properties. Therefore, we should treat those scaling relations carefully.

Our results suggest the possibility of measuring cluster parameters with SZE observation alone. The agreement between our results and those from the literature provides not only confidence for our project but also supports our understanding of galaxy clusters. The upcoming expanded AMiBA with higher sensitivity and better resolution will significantly improve the constraints on these cluster properties. In addition, an improved determination of the - space structure of the clusters directly from AMiBA will make it possible to measure the properties of clusters which currently do not have good X-ray data. The ability to estimate cluster properties based on SZE data will improve the study of mass distribution at high redshifts. On the other hand, the fact that the assumptions of cluster mass and temperature profiles significantly bias the estimations of scaling relations should be also noticed and treated carefully.

We thank the Ministry of Education, the National Science Council (NSC), and the Academia Sinica, Taiwan, for their funding and supporting of AMiBA project. YWL thank the AMiBA team for their guiding, supporting, hard working, and helpful discussions. We are grateful for computing support from the National Center for High-Performance Computing, Taiwan. This work is also supported by National Center for Theoretical Science, and Center for Theoretical Sciences, National Taiwan University for J.H.P. Wu. Support from the STFC for M. Birkinshaw is also acknowledged.

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