II. PERTURBATION THEORY WHEN DE INTERACTS WITH DM In this section, we go over the main results on the linear perturbation theory when DE interacts with DM. The detailed descriptions can be found in [8, 19]. In the spatially flat Friedmann-Robertson-Walker(FRW) background, if there is interaction between DE and DM, neither of them can evolve independently. The (non)conservation equation are where the subscript “m” denotes the cold DM, “d” denotes the DE and Q describes the interaction between them. Since we know neither the physics of DM nor that of DE at the present moment, we cannot write out the precise form of the interaction between them from first principles (see [33] for recent attempts). One has to specify the interaction either from the outset[3], or determine it from phenomenological requirements [5, 18]. For the sake of generality, we consider the phenomenological description of the interaction between DE and DM in the comoving frame [8, 19] where H is the global expansion rate in the conformal time, ξ$_{1}$, ξ$_{2}$ are small dimensionless constants and T is the transpose of the matrix. The perturbation equations for DM and DE in the subhorizon scale have been derived in [19], having the form where the prime denotes the derivative with respect to the conformal time and the gauge-invariant quantities Δ$_{m}$ , Δ$_{d}$ represent the density contrast Δ$_{m}$ ≈ δρ$_{m}$/ρ$_{m}$ = δ$_{m}$ , Δ$_{d}$ ≈ δρ$_{d}$/ρ$_{d}$ = δ$_{d}$ . The perturbed gauge invariant couplings are expressed as [8, 19], 4 a $^{2}$δQ0 I m ρ$_{m}$ ≈ 3 H ( ξ$_{1}$ Δ$_{m}$ + ξ$_{2}$ Δ$_{d}$ /r ) , Δ $^{′}$m = − kV$_{m}$ − Δ$_{m}$ a $^{2}$Q0 m ρ$_{m}$ + a $^{2}$δQ0 I m ρ$_{m}$ , V ′ m = −H V$_{m}$ + k Ψ − a $^{2}$Q0 m ρ$_{m}$ V$_{m}$ + a $^{2}$δQ I pm ρ$_{m}$ ; (3) Δ $^{′}$d = 3 H ( w − C 2 $_{e}$)Δ$_{d}$ − k (1 + w ) V$_{d}$ − Δ$_{d}$ a $^{2}$Q0 d ρ$_{d}$ + a $^{2}$δQ0 I d ρ$_{d}$ , V ′ d + H (1 − 3 w ) V$_{d}$ = kC 2 e 1 + w Δ$_{d}$ + C 2 a 1 + w ρ $^{′}$d ρ$_{d}$ V$_{d}$ + k Ψ − w ′ 1 + w V$_{d}$ − a $^{2}$Q0 d ρ$_{d}$ V$_{d}$ + a $^{2}$δQ I pd (1 + w ) ρ$_{d}$ ; (4) Q ν m = [ 3 H a 2 ( ξ$_{1}$ρ$_{m}$ + ξ$_{2}$ρ$_{d}$ ) , 0 , 0 , 0 ] T Q ν d = [ − 3 H a 2 ( ξ$_{1}$ρ$_{m}$ + ξ$_{2}$ρ$_{d}$ ) , 0 , 0 , 0 ] T , (2) ρ $^{′}$m + 3 H ρ$_{m}$ = a $^{2}$Q0 $_{m}$, ρ $^{′}$d + 3 H (1 + w ) ρ$_{d}$ = a $^{2}$Q0 $_{d}$, (1)