8 Discussion and conclusions

Since many galaxy clusters with available SZ measurements show also the presence of an extended radio halo and other non-thermal phenomena, it is relevant to understand quantitatively the relevance of the non-thermal SZ effect in galaxy clusters and to disentangle it from the true thermal SZ effect which is extensively used for cosmological studies. It is also relevant to use the limits on the non-thermal SZ effect to put constraints on the spectrum of the non-thermal population which is responsible for the high-energy phenomena occurring in galaxy clusters. This proves to be a powerful and complementary analysis of the specific studies of non-thermal phenomena carried out at other wavelenghts.

In such a context, we have shown in this paper that the spectra of the thermal and non-thermal SZ effects are distinctly different and that the overall spectrum of the SZ effects measures the energy densities in the thermal and in the non-thermal electron populations, separately. We have calculated here the spectral shape of the non-thermal SZ effect in the Thomson limit using an exact approach which makes use of the full relativistic formalism and goes beyond the Kompaneets and the single scattering approximation. Moreover we evaluated, for the first time in a consistent way, the total SZ effect arising from a combination of different electron populations residing in the same galaxy cluster: we considered specifically the cases of i) a thermal plus a non-thermal electron populations and ii) two thermal populations with different temperatures and densities. Such a derivation can be obtained, nonetheless, for an arbitrary set of different electron populations.

We have shown in our approach that the spectral distortion of the CMB due to both the thermal and the non-thermal SZ effect can be written in a general, single form which depends separately on the Comptonization parameter and on the spectral shape of the specific SZ effect. Such an expression is valid for both thermal and non-thermal SZ effects produced by single electron populations as well as for the overall SZ effect produced by a combination of different electron populations. We have also shown that the spectral dependence of the SZ effect, contained in the function $\tilde{g}(x)$, cannot be separated from the dependence on the cluster parameters even at the first order approximation in $\tau$, where $\tilde{g}(x)$ depends on the cluster (effective) temperature. At higher-order approximation in $\tau$, $\tilde{g}(x)$ depends, moreover, also on $\tau$. In the exact approach, the spectral shape of the SZ effect depends strongly on the cluster parameters themselves and specifically from the pressure and the optical depth of the considered electron population.

We have calculated in detail the precision at which each approximation order in $\tau$ reproduces the exact form of the SZ effect: we have shown that the third order approximation in $\tau$ yields a precision $\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$<$ }\ }$1% at each interesting frequency for both the non-thermal and the thermal SZ effect.

We have shown that the main spectral differences for the non-thermal SZ effect with respect to the thermal SZ effect are a maximum of the spectral shape which is moved to higher and higher frequencies for increasing values of the lower cutoff in the momentum distribution, p₁. As a consequence, also the null of the non-thermal SZ effect, x₀, is moved to higher frequencies for increasing values of p₁. Since the pressure $P_{\rm rel}$ of the non-thermal electron distribution depends on p₁, the position of x₀ decreases with increasing values of $P_{\rm rel}$ for a fixed normalization of the non-thermal electron density.

We have shown in this paper that the shape of the exact non-thermal SZ effect is different from the 1st order approximation estimate used so far by different authors (Birkinshaw 1999; Ensslin & Kaiser 2000; Blasi et al. 2000; Shimon & Rephaeli 2002). We have shown that the 3rd order approximation is required to reproduce the exact results within a $\sim$1% precision at all relevant values of x.

We have also shown that the SZ effect from a single non-thermal population in clusters with non-thermal phenomena is not a realistic description of the effect. So, we have derived for the first time a consistent expression for the total SZ effect produced by a combination of a thermal and a non-thermal populations, like is the realistic case in many galaxy clusters. The shape and the amplitude of the total SZ effect is quite different from that suggested in previous papers (Ensslin & Kaiser 2000; Blasi et al. 2000). The main difference in the derivation of the overall SZ effect is that it is not just given by the sum of the separate thermal and non-thermal contributions (as is the case for the first order approximation in $\tau$) but it is given by a properly weighted (according to the optical depths of each separate population) combination of the thermal and non-thermal contributions, as discussed in Sect. 4. Also, we considered in our approach the effect of multiple scattering and higher (n>1) order approximation terms in the analytical derivation of the overall SZ effect. Moreover, we also discussed not only the case of i) a single power-law electron spectrum (as was done also by the previous authors) but also the more consistent case of a ii) double power-law electron spectrum which is able to fit both the radio halo and the hard X-ray spectra in clusters. We showed that, while in the case i) low-energy electrons are the dominant source of non-thermal Compton scattering and produce an increase of SZ signal in the region $x \ \raise -2.truept\hbox{\rlap{\hbox{$\sim$ }}\raise5.truept \hbox{$>$ }\ }7$ (where the thermal SZ effect has its maximum), in the case ii) high-energy electrons are the dominant source of non-thermal SZ effect and produce very large shifts of CMB photon frequencies producing a deprivation of photons also in the region of the maximum of the thermal SZ effect ($x \sim 7$).

A relevant result of our analysis shows that the location of the zero of the total SZ effect depends on the pressure ratio $\bar{P} = P_{\rm rel}/P_{\rm th}$ between the relativistic and thermal pressures of the two different electron populations. It increases non-linearly with $\bar{P}$ up to values $x_0 \sim 4.2$ for $\bar{P}$ up to values $\approx$1.5. This is a unique and relevant feature of the non-thermal SZ effect in clusters since it yields, in principle, a direct measure of the total pressure in relativistic non-thermal particles in the cluster atmosphere, an information which is not easily accessible from the study of other non-thermal phenomena like radio halos and/or EUV or hard X-ray emission excesses. Thus, the detailed observations of the frequency shift of x₀ in galaxy clusters provides unambiguously a constraint to the relativistic particle content in the cluster atmosphere. Also, such a measurement is crucial to determine the true amount of kinematic SZ effect in clusters since it is usually estimated from the residual SZ signal at the location of the zero of the thermal (relativistic) SZ effect. Due to the steepness of the SZ spectral shape in the region of the null of the overall SZ effect, the possible additional non-thermal SZ signal must be determined precisely in order to derive reliable limits for the kinematic SZ effect. This fact can severely limit the possibility to measure effectively the kinematic SZ effect in radio-halo clusters. An accurate subtraction of the non-thermal SZ effect needs a specific observational strategy with measurements in at least three frequency ranges $x \sim 2$-3, 3.8-4.5 and $\sim$6-8.

We verified that our approach is consistent with the exact derivation of the monochromatic redistribution function containing both the relativistic and quantum corrections already given by Fargion et al. (1997) and Sazonov & Sunyaev (2000) in the Thomson limit and in the single scattering limit. In the Thomson limit, nonetheless, we have generalized the full derivation of the SZ effect to the case of multiple scattering and to any approximation order in $\tau$.

After the submission of our paper, Shimon & Rephaeli (2002) presented a computation of the SZ effect produced by non-thermal electron populations in a few clusters like Coma and A2199. However, we want to stress that their derivation of the non-thermal SZ effect is still approximate since they consider only the first order approximation in $\tau$ (in which case the total SZ effect is the sum of the thermal and non-thermal effects) and the single scattering limit. We have shown in our paper that such an approximated description is neither complete nor precise since at least the third order approximation calculation in $\tau$ is required to reproduce the exact derivation of the total SZ effect within $\sim$% accuracy level.

We have also generalized our derivation of the total SZ effect in galaxy clusters to the case of a combination of different thermal electron populations (see Sect. 5). Any additional cool IC gas component produces a tightening of the photon redistribution function, an increase in the total optical depth and hence a substantial change in the spectral distortion at the minimum and at the maximum of the SZ effect. The location of the zero of the SZ effect, x₀ also decreases in frequency due to the presence of the cooler component which decreases the pressure ratio P₂/P₁. Thus, the possible detection of an additional cold component in the cluster atmosphere through observations of the total (thermal plus thermal) SZ effect, allow to test the possible thermal origin of the EUV excess observed in several nearby clusters (Lieu et al. 1999, 2000). Our analysis of the cluster A2163 does not indicate a relevant role of the warmer component to the total pressure of the cluster atmosphere, consistently with the present indication of the absence of a strong EUV excess in this cluster. We will address the specific analysis of other clusters elsewhere.

Beyond the relevance of the study of the non-thermal SZ effect as a bias for the cosmologically relevant thermal SZ effect, the non-thermal SZ effect has also a crucial astrophysical relevance as a barometer for the presence of any additional population of electrons with both a non-thermal or a thermal energy spectrum. In fact, the non-thermal SZ effect actually measures the total pressure of the non-thermal electron population and hence yields constraints to its energy spectrum. Analogously, the additional thermal SZ effect produced by a cooler thermal component provides information on its temperature and density. Specifically, we have shown that combining information from the cluster radio-halo observation and non-thermal SZ measurements, one is able to determine the shape and the extension of the relativistic electron spectrum. Using the available observation of one of the best example of radio-halo clusters with multifrequency observations of the SZ effect, A2163, we applied our method and derived a limit on the low-energy cutoff of the relativistic electron spectrum of $E_{\rm min} \sim 50$ MeV, as well as constraints on the momentum spectrum which is required to be flat enough ( $f_{\rm e} \sim p^{-0.5}$) below $p \sim 400$ to avoid destructive feedback effects on the thermal IC gas.

Relativistic electrons with such flat energy spectrum do not produce a relevant extra heating and/or extra X-ray emission with respect to the thermal IC gas in A2163. Also the IC energy losses of such relativistic electrons do not yield a substantial hard X-ray emission, in agreement with the available limit on A2163 obtained from BeppoSAX observations.

We have finally shown that the presence of a non-thermal SZ effect also influence the spatial profile of the total SZ effect of a typical radio-halo cluster. In fact, the region occupied by the radio halo shows an increment (decrement) of the signal at frequencies near the minimum (maximum) of the SZ effect. This happens as a consequence of the difference in the spectral shapes of the non-thermal SZ effect with respect to the thermal one.

The specific spectral and spatial features of the non-thermal SZ effect makes it possible to detect it through a multifrequency observation with high sensitivity and narrow-band detectors. The optimal observational strategy is to observe galaxy clusters in the frequency range $x \sim 2$-8 where the peculiar spectral features allow clearly to disentangle the non-thermal SZ effect from the thermal one. The PLANCK surveyor experiment has the capabilities to detect and map the non-thermal SZ effect in a large number of nearby radio-halo clusters. However, dedicated experiment with high sensitivity and narrow band spectral coverage are also adequate to detect the non-thermal SZ effect in radio-halo galaxy clusters.

Acknowledgements

We thank the referee, N. Itoh, for several useful comments and suggestions which allow us to improve the presentation of our results.