6 no anisotropic stress. As a result, Q and η become scale- dependent and do approach unity on scales smaller than λ$_{σ}$ , see Fig. 2. In addition to this case, scale-dependent Q and η are generated by g$_{P}$ = 0 or g$_{σ}$ = 0, as the scale-dependent effects from the pressure perturbation or anisotropic stress cause the density contrast to evolve in a scale-dependent way. We leave the investigation of these effects for future work. V. INTERACTING DARK ENERGY Dark matter is currently only detected via its grav- itational effects, and there is an unavoidable degener- acy between dark matter and dark energy within GR. There could be a hidden non-gravitational coupling be- tween dark matter and dark energy. In these interacting dark energy (IDE) models the continuity equations for cold dark matter and dark energy are modified by the coupling C such that ˙ ρ$_{c}$ + 3 Hρ$_{c}$ = −C , (34) ˙ ρ$_{de}$ + 3 H (1 + w$_{de}$ ) ρ$_{de}$ = C . (35) where w$_{de}$ = P$_{de}$/ρ$_{de}$ . C is the rate of the energy transfer, so that C > 0( < 0) implies that energy is transferred from dark matter to dark energy (from dark energy to dark matter). In the following, we consider two types of interactions and identify Q and η in these models. In this section, we ignore the clustering of dark energy studied in the previous section assuming that the sound speed is one and there is no anisotropic stress. Also we assume baryons are not coupled to dark energy as there are strong constraints from laboratory tests. First we consider the interaction given by C = Γ$_{c}$ ρ$_{c}$ . A covariant form of the interaction can be found in [48–50] where it was shown that for this form of interaction there are no modifications to the dynamics of quasi-static per- turbations. In fact, in the Newtonian regime the only sig- nal of the dark sector interaction in structure formation to linear order is via the modification of the background expansion history. modes to relax to their late-time asymptotic behaviour after entering the horizon. Q is enhanced for smaller c 2 $_{P}$. The anisotropic stress that is needed to cancel the pressure support, σ$_{de}$ = σ 0 $_{de}$, leads to η < 1 as can be directly seen from Eq. (24). If c 2 P = 0 there is no anisotropic stress, but also there is no scale-dependent suppression of the source term in the growth equation for dark energy perturbations Eq. (31). Therefore dark energy perturbations cluster on all scales like dark matter does. Though the growth of dark energy perturbations is suppressed compared with dark mat- ter perturbations because of a larger Hubble drag and a scale-independent suppression of the source term. A. Model I: C = Γ c ρ c The energy conservation and the Euler equation for density and velocity perturbations in the dark matter and the Poisson equation are given by These equations have precisely the same form as those for the non-interacting case. ˙ δ$_{c}$ + θ$_{c}$ a = 0 , (36) ˙ θ$_{c}$ + Hθ$_{c}$ + k 2 a Φ = 0 , (37) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$( ρ$_{c}$δ$_{c}$ + ρ$_{b}$δ$_{b}$ ) . (38) which acts as a reduced sound horizon such that dark energy only clusters on scales larger than λ$_{σ}$ (as opposed to the scale-invariant case where dark energy clusters on all scales equally.) A slight departure of f$_{σ}$ from unity therefore effectively decreases the sound horizon as com- pared to the sDE. Therefore we expect dark energy to cluster on smaller scales than in the case where there is If the anisotropic stress does not exactly cancel the scale-dependent pressure support in the dark energy growth equation (27), i.e. f$_{σ}$ = 1, then the growth be- comes scale-dependent. The term ∝ k 2 on the rhs of the dark energy growth equation becomes important on quasi-static scales. For f$_{σ}$ > 1 it acts as an instability and δ$_{de}$ would blow up on small scales. For f$_{σ}$ < 1 we can define the scale λ$_{σ}$ ≡ 2 $_{πc$_{P}$}$√$_{1}$ − f$_{σ}$ aH (33) FIG. 2: The scale-dependence of Q and η is shown for different values of f σ . We include the limiting cases of no anisotropic stress ( f σ = 0, dash-dotted) and scale-independent growth of dark energy perturbations ( f σ = 1, dotted). Here we set w de = − 0 . 8 and c 2 P = 1 as in the sDE reference. 1 1.02 1.04 1.06 Q 0.001 0.01 0.1 0.85 0.9 0.95 1 k [h/Mpc] f =1 f =0.9995 f =0.995 f =0