2 gate these functions for different cases: In section III the Brans-Dicke (BD) theory is used to characterise modified gravity models on sub-horizon scales. In section IV we consider clustering dark energy (cDE) models and in sec- tion V interacting dark energy (IDE) models are studied. In section VI we show that these models have distinct paths in the plane of the modified growth parameters µ and Σ. We conclude in section VII. Any deviation from our standard cosmological model sDE will lead to modifications of the background evo- lution and structure growth. Let us fix the background using the sDE reference model and, subsequently, param- eterise the modified growth in terms of the functions Q and η such that these become trivial ( Q = η = 1) in sDE. We assume a flat Friedmann-Robertson-Walker (FRW) background metric. In the sDE model the Friedmann equation defines the evolution of the Hubble parameter, H , in terms of the energy content of the Universe We introduced the label “s” to mark the quantities given in the sDE model. In any other model we adjust the parameters such that H evolves in the same way as in the sDE reference model. Next, we consider linear perturbations around the flat FRW background described in the Newtonian gauge: We will use a dot to denote the derivative with respect to cosmic time, t . In GR the Newtonian potential Ψ and the curvature perturbation Φ are related to the matter content via the linearised Einstein equations. In sDE these yield two constraint equations. Firstly, because there is no anisotropic stress at late times the two metric potentials coincide, Ψ s = − Φ $^{s}$, and secondly, the Pois- son equation relates Φ to the comoving matter density perturbation Δ$_{m}$: For a general fluid ( i ) the comoving density perturbation is defined as in terms of the density contrast δ$_{i}$ and the velocity diver- gence θ$_{i}$ in the Newtonian gauge. The comoving density perturbation Δ$_{m}$ is useful as it is the over-density mea- sured by observers comoving with the matter-flow. In any other GR dark energy or MG model the Poisson equation is modified. This departure from sDE in the Poisson equation is parameterised by the function Q ( a, k ) such that holds with Δ$_{m}$ being the comoving matter perturbation as measured in the modified model (with the same back- ground evolution as sDE) and ρ s m that scales in the usual way with time ( ∝ a − $^{3}$). Because Φ and Δ$_{m}$ are gauge- invariant variables this is a gauge-invariant definition of Q . In MG models, Q represents the mass screening ef- fect due to a local modification of gravity, i.e. an effective modification of Newton’s constant. In exotic dark energy models within GR it measures the additional clustering due to a modification of the background evolution of the energy densities and different evolution of the perturba- tions. In order to fully describe the modified growth of per- turbations, the other constraint equation of GR needs to be parameterised as well. The relation between Φ and Ψ is altered such that is not necessarily trivial. As compared to sDE in GR, where Φ / Ψ = − 1, a non-trivial η effectively adds an anisotropic stress component at linear level. Measuring Φ and therefore Q directly from the galaxy distribution is difficult due to galaxy bias. Therefore, we will also present our results in terms of two re-defined functions that are more directly connected to unbiased observables. First, we notice that weak lensing and the late ISW effect are sensitive to (Φ − Ψ) / 2 and therefore it is convenient to use the combination while peculiar velocities are responding to Ψ and, thus, they determine Note that the growth of matter perturbations is also de- termined by the Newtonian potential, and hence by µ . Therefore, these transformed functions are more closely related to observations. In the following sections we will study the evolution of matter fluctuations inside the Hubble horizon. Then we can use the quasi-static approximation and neglect the time derivatives of the perturbed quantities compared with the spatial derivatives. Furthermore, the comoving density perturbation and the Newtonian gauge density contrast are approximately equal, Δ$_{i}$ ≃ δ$_{i}$ , which means we can neglect the contribution from the velocity po- tentials to the Poisson equation. However, this is not a perfect approximation when considering dark energy perturbations as we will discuss in Sec IV. II. PARAMETERISATION H 2 = 8 πG$_{N}$ 3 ( ρ s m + ρ s $_{de}$) . (1) ds 2 = − (1 + 2Ψ) dt 2 + a $^{2}$(1 + 2Φ) δ$_{ij}$dx $^{i}$dxj . (2) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$Q ( a, k ) ρ s $_{m}$Δ$_{m}$ , (5) η ≡ − Φ Ψ (6) Σ ≡ Q 2 ( 1 + 1 η ) , (7) µ ≡ Q η . (8) k $^{2}$Φs = 4 πG$_{N}$ a $^{2}$ρs $_{m}$Δ s $_{m}$. (3) Δ$_{i}$ ≡ δ$_{i}$ − ˙ ρ$_{i}$ ρ$_{i}$ θ$_{i}$ k 2 , (4)