Dynamics of a dark matter field with a quartic self-interaction potential

Abstract

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 $|\ensuremath{\delta}\ensuremath{\rho}/\ensuremath{\rho}|\ensuremath{\ll}1$ and coherence length L, the energy density fluctuations behave like acoustic oscillations in an ideal relativistic fluid for a time on the order of $L/|\ensuremath{\delta}\ensuremath{\rho}/\ensuremath{\rho}|.$ 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.