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We study non-linear structure formation in the presence of dark energy. The influence of dark energy on the growth of large-scale cosmological structures is exerted both through its background effect on the expansion rate, and through its perturbations. In order to compute the rate of formation of massive objects we employ the spherical collapse formalism, which we generalize to include fluids with pressure. We show that the resulting non-linear evolution equations are identical to the ones obtained in the pseudo-Newtonian approach to cosmological perturbations, in the regime where an equation of state serves to describe both the background pressure relative to density, and the pressure perturbations relative to the density perturbations. We then consider a wide range of constant and time-dependent equations of state (including phantom models) parametrized in a standard way, and study their impact on the non-linear growth of structure. The main effect is the formation of dark energy structure associated with the dark matter halo: non-phantom equations of state induce the formation of a dark energy halo, damping the growth of structures; phantom models, on the other hand, generate dark energy voids, enhancing structure growth. Finally, we employ the Press–Schechter formalism to compute how dark energy affects the number of massive objects as a function of redshift (number counts).

We study the impact of Early Dark Energy fluctuations in the linear and nonlinear regimes of structure formation. In these models the energy density of dark energy is non-negligible at high redshifts and the fluctuations in the dark energy component can have the same order of magnitude of dark matter fluctuations. Since two basic approximations usually taken in the standard scenario of quintessence models, that both dark energy density during the matter dominated period and dark energy fluctuations on small scales are negligible, are not valid in such models, we first study approximate analytical solutions for dark matter and dark energy perturbations in the linear regime. This study is helpful to find consistent initial conditions for the system of equations and to analytically understand the effects of Early Dark Energy and its fluctuations, which are also verified numerically. In the linear regime we compute the matter growth and variation of the gravitational potential associated with the Integrated Sachs-Wolf effect, showing that these observables present important modifications due to Early Dark Energy fluctuations, though making them more similar to ΛCDM model. We also make use of the Spherical Collapse model to study the influence of Early Dark Energy fluctuations in the nonlinear regime of structure formation, especially on δ c parameter, and their contribution to the halo mass, which we show can be of the order of 10%. We finally compute how the number density of halos is modified in comparison to ΛCDM model and address the problem of how to correct the mass function in order to take into account the contribution of clustered dark energy. We conclude that the inhomogeneous Early Dark Energy models are more similar to ΛCDM model than its homogeneous counterparts.

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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, δ c , and on the virial overdensity, ∆ V . The evaluation of their effects on δ 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 δ c and ∆ 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 ΛCDM model and the mass correction due to the contribution of the dark energy component has negligible effects on the overall number of structures.