  The Multi-coupled Dark Energy (McDE) scenario has been recently proposed as a
specific example of a cosmological model characterized by a non-standard
physics of the dark sector of the universe that nevertheless gives an expansion
history which does not significantly differ from the one of the standard
$\Lambda $CDM model. In this work, we present the first constraints on the McDE
scenario obtained by comparing the predicted evolution of linear density
perturbations with a large compilation of recent data sets for the growth rate
$f\sigma_{8}$, including 6dFGS, LRG, BOSS, WiggleZ and VIPERS. Confirming
qualitative expectations, growth rate data provide much tighter bounds on the
model parameters as compared to the extremely loose bounds that can be obtained
when only the background expansion history is considered. In particular, the
$95\%$ confidence level on the coupling strength $|\beta |$ is reduced from
$|\beta |\leq 83$ (background constraints only) to $|\beta |\leq 0.88$
(background and linear perturbation constraints). We also investigate how these
constraints further improve when using data from future wide-field surveys such
as supernova data from LSST and growth rate data from Euclid-type missions. In
this case the $95\%$ confidence level on the coupling further reduce to $|\beta
|\leq 0.85$. Such constraints are in any case still consistent with a scalar
fifth-force of gravitational strength, and we foresee that tighter bounds might
be possibly obtained from the investigation of nonlinear structure formation in
McDE cosmologies.[Abridged]

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology