  We study the nonlinear growth of cosmic structure in different dark energy
models, using large volume N-body simulations. We consider a range of
quintessence models which feature both rapidly and slowly varying dark energy
equations of state, and compare the growth of structure to that in a universe
with a cosmological constant. The adoption of a quintessence model changes the
expansion history of the universe, the form of the linear theory power spectrum
and can alter key observables, such as the horizon scale and the distance to
last scattering. We incorporate these effects into our simulations in stages to
isolate the impact of each on the growth of structure. The difference in
structure formation can be explained to first order by the difference in growth
factor at a given epoch; this scaling also accounts for the nonlinear growth at
the 15% level. We find that quintessence models that are different from
$\Lambda$CDM both today and at high redshifts $(z \sim 1000)$ and which feature
late $(z<2)$, rapid transitions in the equation of state, can have identical
baryonic acoustic oscillation (BAO) peak positions to those in $\Lambda$CDM. We
find that these models have higher abundances of dark matter haloes at $z>0$
compared to $\Lambda$CDM and so measurements of the mass function should allow
us to distinguish these quintessence models from a cosmological constant.
However, we find that a second class of quintessence models, whose equation of
state makes an early $(z>2)$ rapid transition to $w=-1$, cannot be
distinguished from $\Lambda$CDM using measurements of the mass function or the
BAO, even if these models have non-negligible amounts of dark energy at early
times.
