A New Angle on Gravitational Clustering
Román Scoccimarro · 2000
Abstract. We describe a new approach to gravitational instability in large-scale structure, where the equations of motion are written and solved as in field theory in terms of Feynman diagrams. The basic objects of interest are the propagator (which propagates solutions forward in time), the vertex (which describes non-linear interactions between waves) and a source with prescribed statistics which describes the effect of initial conditions. We show that loop corrections renormalize these quantities, and discuss applications of this formalism to a better understanding of gravitational instability and to improving non-linear perturbation theory in the transition to the non-linear regime. We also consider the role of vorticity creation due to shell-crossing and show using N-body simulations that at small (virialized) scales the velocity field reaches equipartition, i.e. the vorticity power spectrum is about twice the divergence power spectrum. 1. Standard Formulation of Gravitational Instability Assuming the initial velocity field is irrotational, gravitational instability can be described completely in terms of the density field and the velocity divergence, θ ≡ ∇ ·v. Defining the conformal time τ = ∫ dt/a and the conformal expansion rate H ≡ dln a/dτ, the equations of motion in Fourier space become ∂ ˜ δ(k) ∂τ θ(k) = − ∂ ˜ θ(k) ∂τ + H ˜ θ(k) + 3 2 ΩH2 ˜ δ(k) = − where [δD] = δD(k − k12), k is a comoving wave number, and d 3 k1d 3 k2[δD]α(k, k1) ˜ θ(k1) ˜ δ(k2), (1) d 3 k1d 3 k2[δD]β(k1, k2) ˜ θ(k1) ˜ θ(k2), (2) α(k, k1) ≡ k · k1 k 2 1, β(k1, k2) ≡ k2 (k1 · k2)