Gravitational instability of scalar fields and formation of primordial black holes

The gravitational instability of a spatially uniform state of a relativistic scalar field on a time-independent background is considered. The instability is demonstrated to be similar to the well-known Jeans instability of dust-like non-relativistic matter. The consideration is generalized to incorporate ‘hydrodynamics’, i.e. self-interaction of the field. It is demonstrated that ‘hydrodynamic’ effects may drastically alter the character of the instability; in particular, the dependence of the instability growth rate on perturbation wavenumber may become non-monotone. The non-linear stage of instability development is analysed half-quantitatively. It is demonstrated that, the ‘hydrodynamics’ being stabilizing, the instability growth may result in establishing stable spatially periodic structures in the one-dimensional situation. The results obtained are applied to estimating the probability of primordial black hole formation from perturbations growing due to gravitational instability in a universe dominated by a scalar axion field.