When Quantum Corrections Alter the Predictions of Classical Field Theory for Scalar Field Dark Matter

We investigate the timescale on which quantum corrections alter the predictions of classical field theory for scalar field dark matter. This is accomplished by including second order terms in the evolution proportional to the covariance of the field operators. When this covariance is no longer small compared to the mean field value, we say that the system has reached the ``quantum breaktime" and the predictions of classical field theory will begin to differ from those of the full quantum theory. While holding the classical field theory evolution fixed, we determine the change of the quantum breaktime as total occupation number is increased. This provides a novel numerical estimation of the breaktime based at high occupations $n_{tot}$ and mode number $N=256$. We study the collapse of a sinusoidal overdensity in a single spatial dimension. We find that the breaktime scales as $\log(n_{tot})$ prior to shell crossing and then then as a powerlaw following the collapse. If we assume that the collapsing phase is representative of halos undergoing nonlinear growth, this implies that the quantum breaktime of typical systems may be as large as $\sim 30$ of dynamical times even at occupations of $n_{tot}\sim 10^{100}$.