  We study the impact of Early Dark Energy fluctuations in the linear and
non-linear 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 $\Lambda$CDM 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 $\delta_c$ 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 $\Lambda$CDM 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 $\Lambda$CDM model than its homogeneous
counterparts.
