  It has been recently shown that the exponential growth rate of a population
of bacterial cells starting from a single cell shows transient oscillations due
to early synchronized bursts of division. These oscillations are enhanced by
cell size regulation and contain information about single-cell growth
statistics. Here, we report a phase transition in these oscillations as a
function of growth rate variability. Below the transition point, these
oscillations become asymptotically deterministic and can be measured
experimentally, while above the transition point, the stochasticity in
population growth dominates the oscillations and masks all the information
about single cell growth statistics. The analytically calculated transition
point, which roughly corresponds to $13\%$ variability in single-cell growth
rate, falls within physiologically relevant parameters. Additionally, we show
that the oscillations can stochastically emerge even when the initial state
contains multiple cells with out-of-phase division cycles. We show that the
amplitude and the phase of these oscillations are stochastic and would vary
across repeated measurements with the same initial conditions. We provide
analytic expressions as well as numerical estimates for the typical oscillation
amplitude and the number of generations before the amplitude falls below a
given measurement threshold for E. coli in multiple growth conditions.
