Non-Relativistic Formation of Scalar Clumps as a Candidate for Dark Matter

We propose a new mechanism for the formation of dark matter clumps in the radiation era. We assume that a light scalar field oscillates harmonically. We include self-interactions and consider the nonrelativistic regime, where the scalar dynamics are akin to the one of a fluid whose pressure depends on both quantum and self-interaction effects. When the squared speed of sound of the scalar fluid becomes negative, an instability arises and the fluctuations of the scalar energy-density field start growing and eventually reach the nonlinear regime, where clumps form. Subsequently, the clumps aggregate and reach a universal regime. We apply this mechanism to a model with a negative quartic term stabilised by a positive self-interaction of order six, and then to axion monodromy, where a subdominant cosine potential corrects a mass term. In the first case, the speed of sound squared becomes negative when the quartic term dominates, leading to a tachyonic instability. For axion monodromy, the instability starts very slowly. First, the density perturbations perform acoustic oscillations due to the quantum pressure. Eventually, they start growing exponentially due to a parametric resonance. The mass and size of the clumps depend crucially on their formation mechanisms. When a tachyonic phase takes place, the core density of the clumps is uniquely determined by the energy density at the beginning of the instability. For axion monodromy, the core density scales with the soliton mass and radius. This difference comes from the role that the quantum pressure plays in the parametric resonance and in balancing the self-interactions and gravity to form static scalar solitons. In both scenarios, we find that the formation redshift of the scalar clumps can span a large range in the radiation era from $10^{-2}$ to $10^4 \, {\rm GeV}$ and the scalar-field mass can be from $10^{-18}$ GeV to $100$ GeV.