Coupled quintessence with a $\Lambda$CDM background: removing the
  $\sigma_8$ tension
  A well-known problem of the $\Lambda$CDM model is the tension between the
relatively high level of clustering, as quantified by the parameter $\sigma_8$,
found in cosmic microwave background experiments and the smaller one obtained
from large-scale observations in the late Universe. In this paper we show that
coupled quintessence, i.e. a single dark energy scalar field conformally
coupled to dark matter through a constant coupling, can solve this problem if
the background is taken to be identical to the $\Lambda$CDM one. We show that
two competing effects arise. On one hand, the additional scalar force is
attractive, and is therefore expected to increase the clustering. On the other,
in order to obtain the same background as $\Lambda$CDM, coupled quintessence
must have a smaller amount of dark matter near the present epoch. We show that
the second effect is dominating today and leads to an overall slower growth.
Comparing to redshift distortion data, we find that coupled quintessence with
$\Lambda$CDM background solves the tension between early and late clustering.
We find for the coupling $\beta$ and for $\sigma_8$ the best fit values
$|\beta| = 0.079^{+ 0.059}_{- 0.067}$ and $\sigma_8 = 0.818^{+0.115}_{-0.088}$.
These values also fit the lensing data from the KiDS-450 survey. We also
estimate that the future missions SKA and Euclid will constrain $\beta$ with an
error of $\pm\, 1.5\times10^{-3}$ and for $\sigma_8$ of $\pm
\,1.8\times10^{-3}$ at $1\sigma$ level.
