Spherical collapse in coupled quintessence with a $\Lambda$CDM
  background
  In this work we study the growth of cold dark matter density perturbations in
the nonlinear regime on a conformally coupled quintessence model in which the
background is designed to mimic a $\Lambda$CDM cosmology. The spherical
collapse of overdense regions is analyzed. We highlight the role of the
coupling on the overall dynamics, trace the evolution of the density contrast
throughout the cosmic history and compute perturbative parameters such as the
critical density contrast. We find that the coupling has the influence of
delaying the collapse due to the slower growth of matter perturbations. We
follow to compute the cluster number counts using the Press-Schechter and
Sheth-Tormen mass functions. In both cases, the transfer of energy between the
dark energy field and dark matter suppresses the number of objects at low
redshifts and enhances the number at high redshifts. Finally, we compute the
expected cluster number counts for the future eROSITA mission and the current
South Pole Telescope survey.
