A GRAVITATIONAL THERMODYNAMIC APPROACH TO PROBE THE PRIMORDIAL SPECTRUM OF COSMOLOGICAL DENSITY-FLUCTUATIONS

Gravitational thermodynamic theory predicts the probability distribution function f(N) for finding N galaxies in a sampling volume V. We compared the theoretical function f(N), which contains a single parameter b, with the results of the N-body simulations emerging from various density perturbation spectra in a flat universe. Originally b is introduced as a global quantity, i.e., independent of the volume size. It is shown, however, that the N-body results match the predicted function f(N) quite well only when the parameter b is regarded as scale-dependent, i.e., b = b(R), where R is a linear size of a sampling volume V. In fact, b(R), obtained by fitting the actual distribution to f(N), turns out to be a useful measure to quantify the present clustering in the universe; in particular, this measure is very sensitive to linear structure on large scales. In this sense, b(R) is complementary to the more conventional two-point correlation function, which is a useful statistic for nonlinear regions. In principle, b(R) can be obtained by fitting the observed distribution of galaxies in two or three dimensions to the theoretical distribution function f(N). Using this procedure, it would be even possible to observationally determine the shape of the primordial spectrum of the density fluctuations.