A coarse-grained field theory for density fluctuations and correlation functions of galaxies and clusters

Abstract

Aims. We aim to investigate the large-scale structure of the universe from a field-theoretical perspective, and to give a unifying understanding of several seemingly unrelated observed features of clusterings of galaxies and clusters, such as the dependence of clustering amplitude on mass and luminosity, the departure of the correlation function ξ(r) from the power law ∞(r o /r) 1.7 the oscillatory behavior of ξcc(r) for clusters on very large scales, and the scaling of r 0 with the mean separation d of clusters. Methods. We present a coarse-grained field theory of density fluctuations for a Newtonian self-gravitating many-body system, apply it to a homogeneous Universe with small density fluctuations, and uniformly treat the clustering of all types of galactic objects in terms of the field of density fluctuations. Results. The Jeans length λ 0 , a unique physical scale for a gravitating system, appears naturally as the characteristic scale underlying the large-scale structure. Under a Gaussian approximation the analytic expressions of ξ(r) and P(k) are obtained. The correlation amplitude is proportional to the galactic mass, ξ(r) ∞ m, and ξ(r) departures from the power law (r o /r) 1.7 around r ∼ 20 h -1 Mpc, and is oscillating over large scales ∼ 100 h -1 Mpc and damped to zero. The spectrum amplitude is inversely proportional to the galactic number density, P(k) ∞ 1/n, and P(k) cc k- 2.2 in the range k≃ (0.05 ∼ 0.3) h Mpc -1 . The scaling r 0 cc d 0.3∼0.5 holds quantitatively. Conclusions. These preliminary analytic results under the Gaussian approximation already qualitatively explain these pronounced features of the observed large-scale structures. Yet on small scales the predicted clustering is insufficient, and the peak of P(k) is too sharp. These deficiencies are due to neglecting higher order nonlinear effects, which need to be studied further.