Dynamics of a Dark Matter Field with a Quartic Self-Interaction Potential

It may prove useful in cosmology to understand the behavior of the energy distribution in a scalar field that interacts only with gravity and with itself by a pure quartic potential, because if such a field existed it would be gravitationally produced, as a squeezed state, during inflation. It is known that the mean energy density in such a field after inflation varies with the expansion of the universe in the same way as radiation. I show that if the field initially is close to homogeneous, with energy density contrast |δρ/ρ| 1 and coherence length L, the energy density fluctuations behave like acoustic oscillations in an ideal relativistic fluid for a time on the order of L/|δρ/ρ|. This ends with the appearance of features that resemble shock waves but interact in a close to elastic way that reversibly disturbs the energy distribution.