content_image/395278.jpg content_image/395297.jpg content_image/395298.jpg content_image/395299.jpg content_image/395292.jpg content_image/395293.jpg content_image/395277.jpg content_image/395283.jpg content_image/395284.jpg content_image/395285.jpg content_image/395291.jpg
<img src='content_image/395278.jpg'>

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 

where we used the fact that δ$_{m}$ ≡ δ$_{b}$ = δ$_{c}$ . Comparing this with the definition of Q in Eq. (5) 

we derive 

In Fig. 3 we plot the parameter Q ( a, k ) from Eq. (41), with different values of the interaction rate Γ$_{c}$. 

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 

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]: 

<img src='content_image/395297.jpg'>

<img src='content_image/395298.jpg'>

<img src='content_image/395299.jpg'>

## B. Model II: C = Γ de ρ de 

<img src='content_image/395292.jpg'>

<img src='content_image/395293.jpg'>

<img src='content_image/395277.jpg'>

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: 

<img src='content_image/395283.jpg'>

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 

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 

<img src='content_image/395284.jpg'>

<img src='content_image/395285.jpg'>

<img src='content_image/395291.jpg'>

