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Our previous analysis indicates that small-scale fluctuations in theintracluster medium (ICM) from cosmological hydrodynamic simulations follow thelognormal distribution. In order to test the lognormal nature of the ICMdirectly against X-ray observations of galaxy clusters, we develop a method ofextracting statistical information about the three-dimensional properties ofthe fluctuations from the two-dimensional X-ray surface brightness.  We first create a set of synthetic clusters with lognormal fluctuations.Performing mock observations of these synthetic clusters, we find that theresulting X-ray surface brightness fluctuations also follow the lognormaldistribution fairly well. Systematic analysis of the synthetic clustersprovides an empirical relation between the density fluctuations and the X-raysurface brightness. We analyze \chandra observations of the galaxy clusterAbell 3667, and find that its X-ray surface brightness fluctuations follow thelognormal distribution. While the lognormal model was originally motivated bycosmological hydrodynamic simulations, this is the first observationalconfirmation of the lognormal signature in a real cluster. Finally we check thesynthetic cluster results against clusters from cosmological hydrodynamicsimulations. As a result of the complex structure exhibited by simulatedclusters, the empirical relation shows large scatter. Nevertheless we are ableto reproduce the true value of the fluctuation amplitude of simulated clusterswithin a factor of two from their X-ray surface brightness alone.  Our current methodology combined with existing observational data is usefulin describing and inferring the statistical properties of the three dimensionalinhomogeneity in galaxy clusters.

Extracting Galaxy Cluster Gas Inhomogeneity from X-ray Surface Brightness: A Statistical Approach and Application to Abell 3667