4 As we want to get a feeling for the time- and scale- dependence of Q and η we take a look at the evolution equations of δ$_{m}$ and δ$_{de}$ without specifying the dark en- ergy pressure and anisotropic stress perturbations. Later we will consider a simple model for the pressure pertur- bation and a specific model for the anisotropic stress of dark energy motivated by the possibility of having scale- invariant growth of dark energy perturbations. From the energy-momentum conservation of the mat- ter and the dark energy we can derive second-order dif- ferential equations for δ$_{m}$ and δ$_{de}$ in the quasi-static ap- proximation where time derivatives of the gravitational potentials are neglected. For the matter component we find the well-known growth equation with the source term as found from the two Einstein con- straint equations above: Here Ω$_{i}$ = Ω$_{i}$( t ) are the fractional energy densities as functions of time. For the general barotropic dark energy fluid perturbations we find the evolution equation ¨ δ$_{m}$ + 2 H ˙ δ$_{m}$ = − k 2 a $_{2}$Ψ , (25) − k 2 a $_{2}$Ψ = 3 2 H 2 { Ω$_{m}$ δ$_{m}$ + Ω$_{de}$ δ$_{de}$ + 3(1 + w$_{de}$ )Ω$_{de}$ σ$_{de}$ } . (26) ¨ δ$_{de}$ + (2 − 6 w$_{de}$ ) H ˙ δ$_{de}$ + 3 H ( δP$_{de}$ ρ$_{de}$ ) . = 3 H ˙ w$_{de}$δ$_{de}$ + 3 [ (2 − 3 w$_{de}$ ) H 2 + ˙ H ] [ w$_{de}$δ$_{de}$ − δP$_{de}$ ρ$_{de}$ ] − (1 + w$_{de}$ ) k 2 a 2 δP$_{de}$ (1 + w$_{de}$ ) ρ$_{de}$ − σ$_{de}$ + Ψ ] . (27) A priori there are no evolution equations for δP$_{de}$ and σ$_{de}$ available. To close the system we need some relation between the pressure and anisotropic stress perturbations and the density perturbation. Formally, cDE can predict any Q ( a, k ) and η ( a, k ). From Eqs. (23) and (24), we can derive δ$_{de}$ and σ$_{de}$ in terms of Q and η . Then substituting them into Eq. (27), we can find a solution for δP$_{de}$ . However, in physical models, δP$_{de}$ and σ$_{de}$ should be determined by micro- physics of the dark energy and it is very unlikely that we have enough degrees of freedom to fine-tune these quantities to get any desirable Q and η . Of course it is a formidable task to find a theoretical prior for dark energy clustering. In the following, we consider a toy model for δP$_{de}$ and σ$_{de}$ to get some idea. δP$_{de}$ ρ$_{de}$ ≡ c 2 $_{P}$δ$_{de}$ + g$_{P}$ k 2 . (28) The pressure perturbation is related to the density per- turbation via the speed of sound. Usually one defines the speed of sound in the rest-frame of the fluid where the pressure perturbations are assumed to be adiabatic: ˆ c 2 s ≡ ( δP$_{de}$/δρ$_{de}$ ) rest . The transformation back into the Newtonian gauge leads to a correction to δP$_{de}$ which is proportional to the velocity potential, θ$_{de}$/k $^{2}$. Here we simply parameterise the pressure perturbation as for cDE models. By combining the two constraints we find that the η parameter here is Q ( a, k ) = 1 + ρ$_{de}$δ$_{de}$ ρ$_{m}$δ$_{m}$ , (23) η = ρ$_{m}$δ$_{m}$ + ρ$_{de}$δ$_{de}$ ρ$_{m}$δ$_{m}$ + ρ$_{de}$δ$_{de}$ + 3(1 + w$_{de}$ ) ρ$_{de}$σ$_{de}$ . (24) Because the evolution of the background energy densities in cDE is not modified with respect to the sDE reference, we have ρ s m = ρ$_{m}$ . So the definition of Q in Eq. (5) directly implies k $^{2}$(Φ + Ψ) = − 12 πG$_{N}$ a $^{2}$(1 + w$_{de}$ ) ρ$_{de}$σ$_{de}$ . (22) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$( ρ$_{m}$δ$_{m}$ + ρ$_{de}$δ$_{de}$ ) (21) The dark energy anisotropic stress, σ$_{de}$ , enters the second Einstein constraint equation as Given the different physical behaviour of matter and dark energy the total density fluctuation will not be a simple linear function of δ$_{m}$ but acquires a contribution from the dark energy. The Poisson equation in cDE mod- els is then given by The dark energy can in principle support long-lived fluctuations [38–46]. In these clustering dark energy (cDE) models the baryons will fall into the potential wells created by both the dark matter and energy. Here we consider a general dark energy fluid with a time-varying barotropic equation of state parameter, w$_{de}$ ≡ P$_{de}$/ρ$_{de}$ , and some scalar anisotropic stress σ$_{de}$ . However, in this section we restrict the dark energy to be minimally cou- pled to gravity and (dark and baryonic) matter, which means that here ρ$_{i}$ = ρ s i for i = m, de ; and we can treat the baryons and the dark matter as a single pressureless perfect fluid. IV. CLUSTERING DARK ENERGY This BD theory contains two well known modified gravity models as special cases: f ( R ) gravity and the DGP braneworld model. Firstly, in f ( R ) gravity we have ω BD = 0 [34]. In this case µ = 4 / 3 for Ma/k ≪ 1 and µ = 1 for Ma/k ≫ 1. In f ( R ), though, G$_{N}$ can be time- dependent, which effectively modifies Σ and µ . However, its dependence on time is strongly constrained by obser- vations and we can safely neglect such an effect. Sec- ondly, in the DGP model we have ω BD = (3 / 2)( β ( t ) − 1) with β ( t ) = 1 − 2 Hr$_{c}$ (1 + ˙ H/ 3 H $^{2}$), where ˙ H is the time derivative of the Hubble parameter and r$_{c}$ ∼ H − 1 0 is the cross-over scale, which is a parameter in this model [33]. Here M = 0 for quasi-static perturbations. We should note that 3 + 2 ω BD < 0 and µ < 1 in DGP indicating a theoretical pathology of the model [37]. no scalar force and GR is recovered, so µ = 1. On the other hand, for r < r$_{M}$ , the effect of the mass can be neglected and we get the same result as for the massless case. Thus, for M = 0, µ becomes scale-dependent.