7 Application to specific galaxy clusters

The non-thermal SZ effect may already have been observed as a part of the SZ signal from some hot galaxy clusters which possess also non thermal phenomena like radio halos or relics.

In the Coma cluster, in fact, Herbig et al. (1995) detected a strong SZ effect at the level of $y \sim 10^{-4}$ from an observation at low frequencies $\nu_{\rm r} \sim 30$ MHz. Since the Coma cluster has a bright and extended radio halo (see, e.g., Deiss et al. 1997), it is well possible that some fraction of the SZ signal might be contributed by the non-thermal SZ effect. However, to disentangle the non-thermal SZ effect from the thermal one a detailed spectral coverage with reasonably good sensitivity in the frequency range from $x \sim 2.5$ up to $x \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }8$ is needed, with particular care to the region $x \sim$3.8-4. At the time of the acceptance of this paper, the available SZ observations for Coma at these relevant frequencies are still lacking and we will discuss this case in more details in a further paper (Colafrancesco & Marchegiani 2002).

In the following, nonetheless, we will discuss specifically the case of A2163 for which there are observations at different frequencies which cover the interesting part of the SZ spectrum and can be used to put constraints on the presence and on the nature of the non-thermal SZ effect in this radio-halo cluster.

7.1 The case of A2163

The cluster A2163 (z =0.203) is one of the hottest ( $k_{\rm B}T_{\rm e} \sim 12.5$ keV) clusters which possesses a giant radio halo (Herbig & Birkinshaw 1994; Feretti et al. 2001) with diameter size of $\sim (2.9 \pm 0.1) h^{-1}_{50}$ Mpc which is centered on the peak of the X-ray emission. The slope of the power-law synchrotron spectrum is $\alpha_{\rm r} \approx 1.6 \pm 0.3$ and has been estimated from radio data at 1.365 and 1.465 GHz (Feretti et al. 2001); this is consistent with the results obtained by Herbig & Birkinshaw (1994) in the frequency range 10 MHz-10 GHz. There is, however, no evidence of hard X-ray excess due to a non-thermal component in the BeppoSAX data of this cluster (Feretti et al. 2001).

A strong SZ effect has been observed in A2163 at different frequencies. In particular the SZ effect spectrum has been observed from BIMA at 28.5 GHz (LaRoque et al. 2002), from DIABOLO at 140 GHz (Desert et al. 1998) and from SuZIE at 140, 218 and 270 GHz (Holzapfel et al. 1997; these data are dust-corrected in LaRoque et al. 2002) thus including and bracketing the null of the thermal SZ effect. Considering a pure thermal SZ effect, the previous data on A2163 are fitted with a Compton parameter $y_{\rm th} = (3.65 \pm 0.40) \times 10^{-4}$ and with the addition of a kinematic SZ effect whose amplitude corresponds to a positive peculiar velocity $V_{\rm p} = 415^{+920}_{-765}$ km s^-1 (Carlstrom et al. 2002) whose large uncertainties, however, makes it consistent with a zero value. Note that, in principle, the addition of a positive-velocity kinematic SZ effect contributes to provide a deeper negative SZ signal in the region between the minimum and the zero of the SZ effect.

Here, we re-analyzed the data on A2163 trying to put constraints on the possible presence of a non-thermal SZ effect by fitting the available data with a combination of a thermal and non-thermal SZ effect.

The thermal population in A2163 has a temperature of $k_{\rm B}T_{\rm e}=12.4\pm 0.5$ keV and a central density of $n_{\rm e,th}\simeq6.82\times 10^{-3}$ cm^-3. The parameters describing the spatial distribution of the IC gas are $r_{\rm c}=0.36 ~ h_{50}^{-1}$ Mpc and $\beta=0.66$ (Elbaz et al. 1995; Markevitch et al. 1996). With these values the optical depth towards the cluster center is $\tau_{\rm0,th}=1.56\times 10^{-2}$ and the central Comptonization parameter is $y_{\rm0,th}=3.80\times 10^{-4}$.

We show in Fig. 27 the relativistic, thermal SZ effect expected for A2163 compared to the available data. The thermal SZ effect fits the data with a $\chi^2=1.71$ and hence it is statistically acceptable. However, we will show in the following that the inclusion of a non-thermal component of the SZ effect is able to improve sensitively the fit to the data.

We consider a non-thermal population with a double power-law spectrum with parameters $\alpha _1=0.5$, $\alpha _2=2.5$, $p_{\rm cr}=400$, $p_2\rightarrow \infty$and we set p₁ and the density $n_{\rm e,rel}(\tilde{p}_1)~$, with $\tilde{p_1}=100$, as a free parameters. We also assume that the spatial distribution of the non-thermal population is similar to that of the thermal population as indicated by the extension of the radio halo in A2163 (see, e.g., Feretti et al. 2001). In our calculations, we fix p₁ and we search for the value of $n_{\rm e,rel}(\tilde{p}_1)~$which minimizes the $\chi ^2$. We show in Table 12 and in Fig. 27 the results of our calculations. We see that the data are better fitted with values of p₁ which correspond to the electrons with $p_1\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }10^2$; we see also that these electrons can produce a detectable non-thermal SZ effect with a value of the pressure ( $P_{\rm rel}/P_{\rm th}~$$\sim0.3$) and the density ratio $\bar{n}\sim270$ which do not imply a strong influence on the dynamical and thermal state of the IC gas. A model in which p₁=100 and $n_{\rm e,rel}(\tilde{p}_1)\approx 2.5 \times 10^{-5}$ cm^-3 best fits the data with a $\chi^2_{\rm min} = 1.0534$, much lower than the $\chi^2_{\rm min}=1.71$ yielded by the single thermal population. The previous parameters point to a non-thermal population which carries a non negligible pressure contribution $P_{\rm rel} \approx 0.29 P_{\rm th}$ and which has a spectrum not extended at momenta lower than $p_{\rm min} \simeq 100$, corresponding to $E_{\rm min} \approx 50$ MeV.

This result does not imply that the fit with a combination of thermal plus non-thermal populations is statistically excluding the single thermal population model. However, our analysis shows that: i) a non-thermal SZ effect is produced by relativistic electrons producing radio halo emission has to be present in A2163. The $\chi ^2$ analysis indicates that its amplitude could be appreciable (see Fig. 27) and corresponds to a pressure in relativistic particles $P_{\rm rel} \approx 0.3 P_{\rm th}$; ii) the possibility of having SZ observations with better precision can offer the possibility to disentangle between the thermal and any non-thermal component of the SZ effect; iii) the detection of a non-thermal SZ effect can set strong constraints on the nature of the non-thermal population and on its feedback on the thermal one.

In these respects, it is appealing that the physical characteristics of the non-thermal population can be constrained through a detailed study of the SZ effect observed in the same galaxy cluster. Once the value of p₁ has been set, the quantity $n_{\rm e,rel}(\tilde{p}_1)~$can be constrained with a quite good precision and vice-versa. This example shows one of the potential uses of the SZ effect to obtain information on the properties of different electronic populations which are residing in the atmospheres of galaxy clusters.

 

 

Table 12: For each value of p₁ are indicated the value of the density $n_{\rm e,rel}( \tilde{p}_1=100)$ which provides the best fit and the corresponding values of pressure ( $\bar{P} = P_{\rm rel}/P_{\rm th}$) and density ( $\bar{n}= n_{\rm e,th}/n_{\rm e,rel}$) ratio with the corresponding value of $\chi ^2$ for the total SZ effect in A2163 produced by a combination of a thermal population and a non-thermal double power-law population.

p1	$n_{\rm e,rel}(\tilde{p}_1)~$(cm-3)	$\bar{P}$	$\bar{n}$	$\chi ^2$
0.1	$1.657\times10^{-5}$	0.25	260	1.0536718
1	$1.664\times10^{-5}$	0.25	266	1.0536183
10	$1.793\times10^{-5}$	0.26	270	1.0534818
100	$2.527\times10^{-5}$	0.29	270	1.0534685
1000	$7.991\times10^{-4}$	0.38	270	1.0534679
10 000	$2.527\times10^{-2}$	0.46	270	1.0534679

Figure 27: Theoretical expectations for the spectrum of the SZ effect in A2163. We show the fit to the available data yielded by a thermal population (solid curve) and the expectations obtained from a combination of thermal and non-thermal populations with p1=100 for a value of the pressure ratio $P_{\rm rel}/P_{\rm th} = 0.29$ (dashed curve), which provides the best fit.

Using our general approach described in the previous sections, we also considered the calculation of the total SZ effect produced by a combination of two thermal electron populations, with the warm population temperature in the range $k_{\rm B}T \sim 0.1$-1 keV and density in the range $n_{\rm e,2} \sim 10^{-2}$-10^-4 cm^-3. We also considered two different spatial distributions of the hot and warm populations: in the first case the two populations have the same spatial extent and in the second one the cooler population is spatially more extended than the hotter one. In this last case we use parameters $r_{\rm c,2}=1.5 ~ r_{\rm c,1}$ and $\beta_2=0.5$.

 

 

Table 13: For each value of $k_{\rm B} T_{\rm e,2}$ are indicated the value of the density n₂ which provides the best fit and the corresponding values of pressure and density ratio with the corresponding value of $\chi ^2$ for the total SZ effect in A2163 produced by a combination of a two thermal populations with the same spatial distribution.

$k_{\rm B} T_{\rm e,2}$ (keV)	$n_{\rm e,2}$ (cm-3)	P2/P1	$n_{\rm e,1}/n_{\rm e,2}$	$\chi ^2$
0.1	$3.922\times10^{-2}$	$4.64\times10^{-2}$	0.17	1.5633934
0.5	$8.508\times10^{-3}$	$5.03\times 10^{-2}$	0.80	1.2395049
1	$4.222\times10^{-3}$	$4.99\times10^{-2}$	1.61	1.2307970

The results of the fit to the data of A2163 are reported in Table 13 and in Fig. 28, and in Table 14 and Fig. 29. The available SZ observations allow also in this case to set constraints on the parameters $n_{\rm e,2}$ and $T_{\rm e,2}$ A best fit which minimizes the $\chi^2_{\rm min}$is obtained for values $k_{\rm B} T_{\rm e,2}=0.5$-1 keV, while the case $k_{\rm B}T_{\rm e,2}=0.1$keV seems to be disfavoured by the data.

Finally we want to remark that, in the set of models we considered in this section, the best fit to the data (with a $\chi^2_{\rm min} \approx 1.05$) is obtained from a combination of thermal and non-thermal populations, a case which seems to be favoured by the present data on A2163.

Figure 28: The spectrum of the SZ effect in A2163 obtained from a combination of two thermal populations with $k_{\rm B} T_{\rm e,1}=12.4$ keV and $k_{\rm B} T_{\rm e,2}=0.5$ keV and with the same spatial distribution. We show the cases of P2/P1: 0 (solid line), $1.77\times 10^{-2}$ (dashed line), $5.03\times 10^{-2}$ (dotted line, which yields the best fit in this case), $5.91\times 10^{-2}$ (dot-dashed line).

 

 

Table 14: For each value of $k_{\rm B} T_{\rm e,2}$ are indicated the value of the density n₂ which provides the best fit and the corresponding values of pressure and density ratio with the corresponding value of $\chi ^2$ for the total SZ effect in A2163 produced by a combination of a two thermal populations with different spatial distributions.

$k_{\rm B} T_{\rm e,2}$ (keV)	$n_{\rm e,2}$ (cm-3)	P2/P1	$n_{\rm e,1}/n_{\rm e,2}$	$\chi ^2$
0.1	$1.624\times10^{-2}$	$1.92\times10^{-2}$	0.42	1.5911710
0.5	$3.531\times10^{-3}$	$2.09\times 10^{-2}$	1.93	1.2504161
1	$1.752\times10^{-3}$	$2.07\times10^{-2}$	3.89	1.2412285

Figure 29: The spectrum of the SZ effect in A2163 obtained from a combination of two thermal populations with $k_{\rm B} T_{\rm e,1}=12.4$ keV and $k_{\rm B} T_{\rm e,2}=0.5$ keV and with different spatial distributions as explained in the text. We show the cases of P2/P1: 0 (solid line), $5.91\times 10^{-3}$ (dashed line), $2.09\times 10^{-2}$ (dotted line, which yields the best fit), $2.96\times 10^{-2}$ (dot-dashed line).

Future SZ observations of A2163 with higher spectral resolution and sensitivity will allow to set stronger constraints on the non-thermal population. In particular, the possibility to measure accurately the frequency location of x₀ could yield direct information on the value of the pressure in relativistic particles, $P_{\rm rel}$, confined in the cluster atmosphere and on their energy distribution.