7 FIG. 3: The time evolution of Q is presented for two different models of IDE. The left-hand side plot corresponds to Model I: C = Γ c ρ c with varying Γ c (1/Mpc). The right-hand side plot corresponds to Model II: C = Γ de ρ de with varying Γ de (1/Mpc). This fact leads to a linear bias between dark matter and baryon over-densities. The source term leads to a modi- fied growth equation of dark matter: However, a non-trivial Q is induced by the non- adiabatic scaling of ρ$_{c}$ such that ρ$_{m}$ ≡ ρ$_{c}$ + ρ$_{b}$ = ρ s m although ρ$_{b}$ = ρ s $_{b}$. The Poisson equation is written as In this model, there is an ambiguity in the definition of Q due to the bias between baryon and dark matter over-densities. The baryon over-densities obey the same growth equation as that for matter perturbations in the sDE reference model ¨ δ$_{c}$ + 2 H ( 1 − Γ$_{de}$ H ¯ ρ$_{de}$ ¯ ρ$_{c}$ ) ˙ δ$_{c}$ = 4 πG { ¯ ρ$_{b}$δ$_{b}$ + [ 1 + 2 3 a Γ$_{de}$ H ¯ ρ$_{de}$ ¯ ρ$_{c}$ × × { 2 − 3 w + Γ$_{de}$ H ( 1 + ¯ ρ$_{de}$ ¯ ρ$_{c}$ )}] ¯ ρ$_{c}$δ$_{c}$ } . (44) ¨ δ$_{b}$ + 2 H ˙ δ$_{b}$ − 4 πG ( ρ$_{b}$δ$_{b}$ + ρ$_{c}$δ$_{c}$ ) = 0 . (45) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$Q ( a, k ) ρ s $_{m}$δ$_{b}$ , (46) Therefore we define the function Q with respect to the baryons such that the bias is absorbed into Q . Hence we have The reason why we chose to define Q in terms of δ$_{b}$ is that we might be able to use the energy conservation equation of the baryons to measure the baryon density contrast from peculiar velocities of galaxies, which are but the energy conservation equation for the dark matter density contrast now acquires a source term due to the coupling with the dark energy [50, 51]: Next, we consider the interaction given by C = Γ$_{de}$ ρ$_{de}$ . A covariant form of the interaction can be found in [35]. The Euler equation for the velocity perturbation is the same as in the previous model Q ( a, k ) = ρ$_{b}$δ$_{b}$ + ρ$_{c}$δ$_{c}$ ρ s $_{m}$δ$_{b}$ . (47) ˙ θ$_{c}$ + Hθ$_{c}$ + k 2 a Φ = 0 , (42) ˙ δ$_{c}$ + θ$_{c}$ a = Γ$_{de}$ ρ$_{de}$ ρ$_{c}$ δ$_{c}$ . (43) B. Model II: C = Γ de ρ de In Fig. 3 we plot the parameter Q ( a, k ) from Eq. (41), with different values of the interaction rate Γ$_{c}$. where we used the fact that δ$_{m}$ ≡ δ$_{b}$ = δ$_{c}$ . Comparing this with the definition of Q in Eq. (5) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$( ρ$_{b}$ + ρ$_{c}$ ) δ$_{m}$ , (39) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$Q ( a, k ) ρ s $_{m}$δ$_{m}$ , (40) Q ( a, k ) = ρ$_{b}$ + ρ$_{c}$ ρ s m . (41) we derive