Renormalization Group Flow and Fragmentation in the Self-Gravitating Thermal Gas

Abstract

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 ϕ(x) and exponential self-interaction. We build up perturbation theory around a space dependent stationary point ϕ₀(r) in a finite size domain σ≤r≤R, (σ << R which is relevant for astrophysical applications (inter stellar medium, galaxy distributions). We compute the correlations of the gravitational potential (ϕ) and of the density and find that they scale; the density correlator scales as r ⁻². A rich structure emerges in the two-point correlators from the ϕ fluctuations around ϕ₀(r). The n-point correlators are explicitly computed to the one loop level. The relevant effective coupling turns out to be λ=4π G m ²/(T R. The renormalization group equations (RGE) for the n-point correlatorare derived and the RG flow for the effective coupling λ(τ), τ=ln(R/σ), explicitly obtained. A novel dependence on τ emerges here. λ(τ) vanishes each time τ approaches discrete values $\tau = \tau _n = 2\pi n/\sqrt 7 - 0,n = 0,1,2, \ldots$. Such RG stable behaviour [λ(τ) decreasing with increasing τ] is here connected with low density self-similar fractal structures fitting one into another. For sizes smaller than the points τ _n , RG unstable behaviour appears which we connect to Jeans' unstable behaviour, growing density and fragmentation. Remarkably, we get a hierarchy of scales and Jeans lengths following the geometric progression $R_n = R_0 e^{2\pi n/\sqrt 7 } = R_0 [10.749087 \ldots ]^n$. A hierarchy of this type is expected for non-spherical geometries, with a ratio different from $e^{2n/\sqrt 7 }$.