Slow Diffusive Gravitational Instability Before Decoupling

Radiative diffusion damps acoustic modes at large comoving wavenumber (k) before decoupling ("Silk damping"). In a simple WKB analysis, neglecting moments of the temperature distribution beyond the quadrupole (the tight-coupling limit), damping appears in the acoustic mode as a term of order ik^2*τ^-1, where τ is the scattering rate per unit conformal time. Although the Jeans instability is stabilized on scales smaller than the adiabatic Jeans length, I show that the medium is linearly unstable to first order in τ^-1 to a slow diffusive mode. At large comoving wavenumber, the characteristic growth rate becomes independent of spatial scale and constant: (t_KH a)^-1 ≈ (128πG/9κ_T *c)(ρ_m /ρ_b ), where a is the scale factor, ρ_m and ρ_b are the matter and baryon energy density, respectively, and κ_T is the Thomson opacity. This is the characteristic timescale for a fluid parcel to radiate away its total thermal energy content at the Eddington limit, analogous to the Kelvin-Helmholz (KH) timescale for a radiation pressure-dominated massive star or the Salpeter timescale for black hole growth. Although this mode grows at all times prior to decoupling and on scales smaller than roughly the horizon, the growth time is long, about 100 times the age of the universe at decoupling. Thus, it modifies the density and temperature perturbations on small scales only at the percent level. The physics of this mode in the tight-coupling limit is already accounted for in the popular codes CMBFAST and CAMB, but is typically neglected in analytic studies of the growth of primordial perturbations. The goal of this work is to clarify the physics of this diffusive instability in the epoch before decoupling, and to emphasize that the universe is formally unstable on scales below the horizon, even in the limit of very large τ. Analogous instabilities that might operate at yet earlier epochs are also mentioned