The self-gravitating thermal gas (non-relativistic particles of mass m at temperature $T)$ is exactly equivalent to a field theory with a single scalar field $\phi \left(\mathbf{x}\right)$ and exponential self-interaction. We build up perturbation theory around a space dependent stationary point ${\phi }_0\mathrm{}\left(r\right)\mathrm{}$ in a finite size domain $\delta <~r<~R\hspace{0.5em}(\delta \ll R),$ which is relevant for astrophysical applications (interstellar medium, galaxy distributions). We compute the correlations of the gravitational potential $\left(\phi \right)$ and of the density and find that they scale; the latter scales as $r^{-2}.$ A rich structure emerges in the two-point correlators from the $\phi$ fluctuations around ${\phi }_0\mathrm{}\left(r\right).$ The n-point correlators are explicitly computed to the one-loop level. The relevant effective coupling turns out to be $\lambda =4\mathrm{}\pi {\mathrm{Gm}}^2/\left(\mathrm{TR}\right).$ The renormalization group (RG) equations for the n-point correlator are derived and the RG flow for the effective coupling $\lambda \left(\tau \right),$ $\tau =\ln (R/\delta ),$ explicitly obtained. A novel dependence on $\tau$ emerges here. $\lambda \left(\tau \right)$ vanishes each time $\tau$ approaches discrete values $\tau ={\tau }_n=2\mathrm{}\pi n/\sqrt{7}-0,$ $n=0,1,2,\dots \hspace{0.5em}.$ Such RG stable behavior $\left[\lambda \right(\tau )$ decreasing with increasing $\tau$] is here connected with low density self-similar fractal structures fitting one into another. For sizes smaller than the points ${\tau }_n,$ RG unstable behavior appears which we connect to the Jeans unstable behavior, growing density and fragmentation. Remarkably, we get a hierarchy of scales and Jeans lengths following the geometric progression $R_n{=R}_0\hspace{0.5em}{e}^{2\pi n/\sqrt{7}}{=R}_0[10.749087\dots {]}^n.$ A hierarchy of this type is expected for non-spherical geometries, with a ratio different from $e^{2\pi /\sqrt{7}}.$

• Received 21 December 1998

DOI:https://doi.org/10.1103/PhysRevD.59.125021