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.