Constraints from growth-rate data on some coupled dark energy models
  mimicking a $\Lambda CDM$ expansion
  The $\Lambda CDM$ expansion could be mimicked by a dark energy coupled to
matter. Then, the equation of state $\bar w$ and coupling $\bar Q$ of this
coupled dark energy could not be constrained by observations of the Hubble
function alone. Also, in this paper, we determine the constraints on two such
coupled dark energy models considering some current and forecast Euclid-like
growth-rate data and assuming the prior on the $\Lambda CDM$ dark matter
density parameter today $\Omega_{m0}=0.295\pm 0.04$. The first model is defined
by a constant equation of state. We find that at $2\sigma$, $\bar
w=-1.02_{-0.22}^{+0.06}$ and the coupling function $\bar Q_0$ today is $\bar
Q_0H_0^{-3}=0.057_{-0.148}^{+0.353}$ with $H_0$ the Hubble constant. The second
model is defined by a varying equation of state $\bar w=\bar w_a-\bar
w_b\ln(1+z)$, with $z$ the redshift and $(\bar w_a,\bar w_b)$, two constants.
We find that at $2\sigma$, $\bar w_a=-0.99_{-0.90}^{+0.17}$, $\bar
w_b=-0.04_{-1.17}^{+0.31}$ and $\bar Q_0H_0^{-3}=0.0002_{-0.18}^{+1.35}$. These
constraints on coupled dark energy agreed with a $\Lambda CDM$ model but are
too poor to discard confidently a coupled dark energy different from vacuum but
mimicking a $\Lambda CDM$ expansion.
