Slow Diffusive Gravitational Instability Before Decoupling

Abstract

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, damping appears in the acoustic mode as a term of order ik^2/(taudot) where taudot 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 (1/taudot) to a slow diffusive mode. At large comoving wavenumber, the characteristic growth rate becomes independent of spatial scale and constant: (t_{KH}a)^-1 ~ (128 pi G/9 kappa_T c)(rho_m/rho_b), where "a" is the scale factor, rho_m and rho_b are the matter and baryon energy density, respectively, and kappa_T is the Thomson opacity. This is the characteristic timescale for a fluid parcel to radiate away its thermal energy content at the Eddington limit, analogous to the Kelvin-Helmholz (KH) time for a massive star or the Salpeter time for black hole growth. Although this mode grows at all times prior to decoupling and on scales smaller than 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 is already accounted for in the popular codes CMBFAST and CAMB, but is typically neglected in analytic studies of the growth of primordial perturbations. This work clarifies the physics of this instability in the epoch before decoupling, and emphasizes that the universe is formally unstable on scales below the horizon, even in the limit of large taudot. Analogous instabilities at yet earlier epochs are also mentioned. (Abridged)