Effects of shear and rotation on the spherical collapse model for
  clustering dark energy
  In the framework of the spherical collapse model we study the influence of
shear and rotation terms for dark matter fluid in clustering dark energy
models. We evaluate, for different equations of state, the effects of these
terms on the linear overdensity threshold parameter, $\delta_{\rm c}$, and on
the virial overdensity, $\Delta_{\rm V}$. The evaluation of their effects on
$\delta_{\rm c}$ allows us to infer the modifications occurring on the mass
function. Due to ambiguities in the definition of the halo mass in the case of
clustering dark energy, we consider two different situations: the first is the
classical one where the mass is of the dark matter halo only, while the second
one is given by the sum of the mass of dark matter and dark energy. As
previously found, the spherical collapse model becomes mass dependant and the
two additional terms oppose to the collapse of the perturbations, especially on
galactic scales, with respect to the spherical non-rotating model, while on
clusters scales the effects of shear and rotation become negligible. The values
for $\delta_{\rm c}$ and $\Delta_{\rm V}$ are higher than the standard
spherical model. Regarding the effects of the additional non-linear terms on
the mass function, we evaluate the number density of halos. As expected, major
differences appear at high masses and redshifts. In particular, quintessence
(phantom) models predict more (less) objects with respect to the $\Lambda$CDM
model and the mass correction due to the contribution of the dark energy
component has negligible effects on the overall number of structures.
