Abstract
{We used the observations carried out by XMM in the COSMOS field over 3.5 yr to study the long term variability of a large sample of active galactic nuclei (AGNs) (638 sources) in a wide range of redshifts (0.1 {\lt} z {\lt} 3.5) and X-ray luminosities (10$^{41}$ {\lt} L $_{0.5-10}$ {\lt}10$^{45.5}$). Both a simple statistical method to assess the significance of variability and the Normalized Excess Variance ($\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$) parameter were used to obtain a quantitative measurement of the variability. Variability is found to be prevalent in most AGNs, whenever we have good statistics to measure it, and no significant differences between type 1 and type 2 AGNs were found. A flat (slope {\ndash}0.23 {\plusmn} 0.03) anti-correlation between $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ and X-ray luminosity is found when all significantly variable sources are considered together. When divided into three redshift bins, the anti-correlation becomes stronger and evolving with z, with higher redshift AGNs being more variable. We prove, however, that this effect is due to the pre-selection of variable sources: when considering all of the sources with an available $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ measurement, the evolution in redshift disappears. For the first time, we were also able to study long term X-ray variability as a function of M $_{BH}$ and Eddington ratio for a large sample of AGNs spanning a wide range of redshifts. An anti-correlation between $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ and M $_{BH}$ is found, with the same slope of anti-correlation between $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ and X-ray luminosity, suggesting that the latter may be a by-product of the former. No clear correlation is found between $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ and the Eddington ratio in our sample. Finally, no correlation is found between the X-ray $\backslash$sigma \^{}$\{$2$\}$\_$\{$rms$\}$ and optical variability. }