Effects of Formation Epoch Distribution on X-Ray Luminosity and Temperature Functions of Galaxy Clusters

We investigate statistical properties of galaxy clusters in the context of a hierarchical clustering scenario, taking into account their formation epoch distribution; this study is motivated by the recent finding by Fujita and Takahara that X-ray clusters form a fundamental plane in which the mass and the formation epoch are regarded as two independent parameters. Using the formalism that discriminates between major mergers and accretion, the epoch of a cluster formation is identified with that of the last major merger. Since tiny mass accretion following formation does not much affect the core structure of clusters, the properties of X-ray emission from clusters are determined by the total mass and density at their formation time. Under these assumptions, we calculate X-ray luminosity and temperature functions of galaxy clusters. We find that the behavior of the luminosity function differs from the model that does not take into account formation epoch distribution; the behavior of the temperature function, however, is not much different. In our model, the luminosity function is shifted to a higher luminosity and shows no significant evolution up to z ~ 1, independent of cosmological models. The clusters are populated on the temperature-luminosity plane, with a finite dispersion. Since the simple scaling model in which the gas temperature is equal to the virial temperature fails to reproduce the observed luminosity-temperature relation, we also consider a model that takes into account the effects of preheating. The preheating model reproduces the observations much more accurately.