Statistical Mechanics of the Cosmological Many‐Body Problem

Abstract

We derive analytical expressions for the grand canonical partition functions of point masses, and of extended masses (e.g., galaxies with halos), which cluster gravitationally in an expanding universe. From the partition functions, we obtain the system's thermodynamic properties, distribution functions (including voids and counts in cells), and moments of distributions such as their skewness and kurtosis. This also provides an analytical calculation of the evolution of the distribution. Our results apply to both linear and nonlinear regimes of clustering. In the limit of point masses, these results reduce exactly to previous results derived from thermodynamics, thus providing a new, more fundamental foundation for the earlier results.

Clustering of galaxies in brane world models

“…It may be noted that a similar analysis has been done for galaxies interacting though the usual Newtonian potential [26]. Here N !…”

“…This divergence occurs due to the assumption that the galaxies are point like objects. However, this divergence can be removed by taking the extended nature of galaxies into account by introducing a softening parameter which takes care of the finite size of each galaxy [22,26,27]. Thus by incorporating the softening parameter the interaction potential energy between galaxies in brane world models can be represented as…”

“…However, we will incorporate an random parameter, which will take into account the fact that galaxies do actually have an intended structure. As we will be dealing with a large number of galaxies, and individual galaxies will be effectively treated as point particles, we can apply the methods of statistical mechanics and calculate the partition function for this system [21,22,[25][26][27]. The strength of interaction will depend effectively on the average density of the universe [21].…”

“…This assumption makes it possible for gravitational clustering to evolve through a sequence of quasi-equilibrium state [22]. In fact, using this assumption the cosmological many body partition function has been calculated [26,28,29]. This was done by using using an ensemble of co-moving cells, and each of these co-moving cells contained galaxies which interacted with each other.…”

“…It was demonstrated that if the size of the cells is smaller than the particle correlation length, then each member of this ensemble is correlated gravitationally with other cells. Furthermore, the correlations within each cells is greater than correlations among different cells, so that extensive is justified as a good approximation [26,30]. In this paper, we will use this formalism for analyzing a system of galaxies interacting through a modified Newtonian potential.…”