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where r = ρ$_{m}$/ρ$_{d}$ . 

It is useful to rewrite these equations in the real space, 

<img src='content_image/325853.jpg'>

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where ¯ x refers to the conformal coordinates. 

Defining σ$_{m}$ = δρ$_{m}$ , σ$_{d}$ = δρ$_{d}$ , and assuming that the EOS of DE is constant w ′ = 0, we can change Eqs.( 5, 6) into, 

<img src='content_image/325849.jpg'>

<img src='content_image/325850.jpg'>

where ∇$_{x}$ = 1 $_{a}$∇$_{¯}$$_{x}$. The dot denotes the derivative with respect to the cosmic time. Ψ indicates the peculiar potential, which can be decomposed into Ψ = ψ$_{m}$ + ψ$_{d}$ , satisfying the Poisson equation [26], 

<img src='content_image/325848.jpg'>

where σ$_{λ}$ represents the inhomogeneous fluctuation field and the subscript ” λ ” denotes DM or DE, respectively. We have included the correction from General Relativity. In a homogeneous and isotropic background < ψ$_{λ}$ > = 0, since < σ$_{λ}$ > = 0. For DE and DM, their peculiar potentials read [26] 

## III. DERIVATION OF THE LAYZER-IRVINE EQUATION 

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In this section we derive the Layer-Irvine equation [38] when there is an interaction between DE and DM. Layzer- Irvine equation describes how a collapsing system reaches a state of dynamical equilibrium in an expanding universe. 

