5 The gamma-ray - X-ray luminosity correlation

To further strengthen our argument presented in the previous sections, we looked for other intrinsic correlations among galaxy clusters and unidentified EGRET sources. Specifically, we correlated the cluster X-ray luminosity, $L_{\rm X}$, with the luminosity of the associated EGRET source under the assumption that it is physically associated to the cluster and, hence, has the same redshift. We derived the gamma-ray luminosity, $L_{\gamma}$, of each EGRET source from the gamma-ray fluxes at E > 100 MeV given in the Third EGRET catalog (Hartman et al. 1999) using the cluster optical redshift given in Table 1. We consider the same EGRET-cluster associations (with the exception of the source 3EGJ0159-3603 associated with the clusters Abell 219S and Abell 2963 because no reliable redshift is available for these clusters) which show the $F({>}100~{\rm MeV}) - S_{1.4}$ correlation analyzed in the previous Sect. 4. The $L_{\gamma } - L_{\rm X}$ correlation shown by the data (see Fig. 10) is fitted by $L_{\gamma} = C L_{\rm X}^D$ with best fit values $C= 0.06 \pm 0.01$ and $D= 0.593 \pm 0.122$ ($1 \sigma$ errors). A similar result, however, is also found for viewing periods P1234 which do not show, in general, the best S/N ratios for the detected EGRET sources: in this last case we found $C= 0.08 \pm 0.02$ and $D= 0.328 \pm 0.120$ ($1 \sigma$ errors). In any case, the $L_{\gamma } - L_{\rm X}$ relation shown by the data is significant at more than the $2.7 \sigma$ confidence level.

Such a $L_{\gamma } - L_{\rm X}$ correlation indicates a connection between the physical status of the cluster ICM, and of its galaxy content, and the overall gamma-ray emissivity of the cluster: such a connection is indeed expected in the viable model for the gamma-ray emission of galaxy clusters. In fact, both the diffuse emission arising from the interaction of relativistic particles with the cluster ICM and the one arising from a superposition of the gamma-ray emission associated with individual galaxies within the cluster predict a relation $L_{\gamma} \sim L_{\rm X}^{a}$ with $a \approx 0.45 {-} 0.85$. Specifically, the cluster gamma-ray luminosity produced by non-thermal electron bremsstrahlung (see, e.g., Longair 1993),

$$L_{\gamma} \propto n_{{\rm e, rel.}} n R^3 ,$$ (1)

that produced by $\pi^0 \to \gamma + \gamma$ in pp collisions (Colafrancesco & Blasi 1998),

$$L_{\gamma} \propto n_{{\rm p, rel.}} n R^2 ,$$ (2)

and the one produced by $\pi^0 \to \gamma + \gamma$ in dark matter annihilation (Colafrancesco & Mele 2001),

$$L_{\gamma} \propto n^2 R^3 ,$$ (3)

naturally correlate with the cluster X-ray luminosity mainly given by thermal bremsstrahlung,

$$L_{\rm X} \propto n^2 T^{1/2} R^3 ,$$ (4)

through their dependence on the ICM particle density, n. Here, the densities of relativistic electrons, $n_{\rm e, rel.}$, and relativistic protons, $n_{\rm p, rel.}$, are decoupled from the ICM density while the dark matter density is proportional to the ICM density. Note that a scaling similar to that in Eq. (3) applies also to the gamma-ray emission arising from the superposition of the cluster galaxies. Using the previous scalings and the observed X-ray luminosity - temperature relation, $L_{\rm X} \sim T^b$ with $b \approx 3$ (see, e.g., Arnaud & Evrard 1999; Wu et al. 1999), a correlation $L_{\gamma} \sim L_{\rm X}^{D}$, with $D \approx 0.45 {-} 0.85$ is predicted by the previous models, in agreement with our results shown in Fig. 10.

Figure 10: The $L_{\gamma } - L_{\rm X}$ correlation shown by the clusters listed with an asterisk in Table 1. The best fit curve (solid line) is shown together with the $1 \sigma$ (green/dark-gray area) and $3 \sigma$ (yellow/pale-gray area) confidence level region for the fitting parameters. The X-ray luminosity, in units of 1044 erg s-1, is given in the 2-10 keV energy range and the gamma-ray luminosity, at E> 100  MeV, of the associated EGRET source is given in units of 1046 erg s-1.