In the simple case of a constant equation of state , redshift distribution of collapsed structures may constrain dark energy models .
Different dark energy models having the same energy density today but different equations of state give quite different number counts .
Moreover , we show that introducing the possibility that dark energy collapses with dark matter ( “ inhomogeneous ” dark energy ) significantly complicates the picture .
We illustrate our results by comparing four dark energy models to the standard \Lambda -model .
We investigate a model with a constant equation of state equal to -0.8 , a phantom energy model and two scalar potentials ( built out of a combination of two exponential terms ) .
Although their equations of state at present are almost indistinguishable from a \Lambda -model , both scalar potentials undergo quite different evolutions at higher redshifts and give different number counts .
We show that phantom dark energy induces opposite departures from the \Lambda -model as compared with the other models considered here .
Finally , we find that inhomogeneous dark energy enhances departures from the \Lambda -model with maximum deviations of about 15 % for both number counts and integrated number counts .
Larger departures from the \Lambda -model are obtained for massive structures which are rare objects making it difficult to statistically distinguish between models .