One of the best motivated hypotheses in cosmology states that most of the matter in the universe is in the form of weakly-interacting massive particles that decoupled early in the history of the universe and cooled adiabatically to an extremely low temperature. Nevertheless, the finite temperature and horizon scales at which these particles decoupled imprint generic signatures on their small-scale density fluctuations. We show that the previously recognized cut-off in the fluctuation power-spectrum due to free-streaming of particles at the thermal speed of decoupling, is supplemented by acoustic oscillations owing to the initial coupling between the cold dark matter (CDM) and the radiation field. The power-spectrum oscillations appear on the scale of the horizon at kinematic decoupling which corresponds to a mass scale of $\sim {10}^{-4}(T_{\mathrm{d}}/10\mathrm{MeV}{)}^{-3}{M}_{\odot }$ for a CDM decoupling temperature $T_{\mathrm{d}}$. The suppression of the power-spectrum on smaller scales by the acoustic oscillations is physically independent from the free-streaming effect, although the two cut-off scales are coincidentally comparable for $T_{\mathrm{d}}\sim 10\mathrm{MeV}$ and a particle mass of $M\sim 100\mathrm{GeV}$. The initial conditions for recent numerical simulations of the earliest and smallest objects to have formed in the universe, need to be modified accordingly. The smallest dark-matter clumps may be detectable through $\gamma$-ray production from particle annihilation, through fluctuations in the event rate of direct detection experiments, or through their tidal gravitational effect on wide orbits of objects near the outer edge of the solar system.

• Received 5 April 2005

DOI:https://doi.org/10.1103/PhysRevD.71.103520