Formation rate of gravitational structures and the cosmic X-ray background radiation

Analytical expressions for the rates of formation and destruction of gravitationally bound systems are derived assuming that they are originated from primordial random-Gaussian density fluctuations. The resulting formulae reproduce the time derivative of the Press-Schechter mass function in a certain limit. Combining a theoretical model for the evolution of structures with the formation rate, we can make various cosmological predictions which are to be compared with observations. As an example to elucidate such applicability, we evaluate the contribution of clusters of galaxies to the cosmic X-ray background radiation. With the {\it COBE} normalization, we find that the significant fraction of the observed soft X-ray background is accounted for by clusters of galaxies in a cold dark matter universe with $\Omega_0 \sim 0.2$, $\lambda_0 =1- \Omega_0$ and $h \sim 0.8$.