3 In both cases Q and η approach unity. For M = 0, there is a unique trajectory in the Q − η plane parameterised by ω BD . As discussed in the previous section, the parameters that are more closely related to observations are Σ and µ ; see the definitions (7) and (8). We find that these parameters are given by Σ( k, a ) = 1 , µ ( k, a ) = 2(2 + ω BD ) + M $^{2}$a$^{2}$/k2 3 + 2 ω BD + M $^{2}$a$^{2}$/k2 . (16) Interestingly, Σ( k, a ) = 1 is the same as in GR. This is because the scalar field couples to the trace of the energy- momentum tensor and it does not couple to the photon. Since Σ parameterises the lensing potential, which in turn determines the geodesics of photons, it is not affected by the scalar field and, thus, Σ = 1 as in GR. On the other hand, µ measures the strength of Newton gravity. In order to understand the result, it is instructive to find a solution for the scalar filed ϕ with a localised matter source δρ$_{m}$ in the Mikowski background. Let us first consider the case with M = 0. Assuming spherical symmetry, the solution for ϕ ( r ) far away from the source in real space is given by where ρ$_{m}$ is the background dark matter energy density, δ$_{m}$ is dark matter density contrast and ϕ is the BD scalar field. In the quasi-static approximation the BD param- eter ω BD and Newton’s constant G$_{N}$ can be any func- tions of time. In the following we assume that G$_{N}$ can be approximated by the one measured in Cavendish-like experiments and the it is constant. As mentioned in the introduction, the large distance modification of gravity, which is necessary to explain the late-time acceleration, generally modifies gravity even on sub-horizon scales due to the introduction of a new scalar degree of freedom. This modification of gravity due to the scalar mode can be described by Brans-Dicke (BD) gravity [23, 24, 36]. In the quasi-static approximation the perturbed modified Einstein equations give III. QUASI-STATIC PERTURBATIONS IN MODIFIED GRAVITY MODELS ϕ ( r ) = − 1 3 + 2 ω BD r$_{g}$ r , (17) k $^{2}$Ψ = − 4 πG$_{N}$ a $^{2}$ρ$_{m}$ δ$_{m}$ + k 2 2 ϕ . (18) where r$_{g}$ ≡ 8 πG$_{N}$ ∫ dr $^{′}$r′ $^{2}$δρ$_{m}$ is the gravitational length of the source. If 3 + 2 ω BD > 0, the scalar field mediates an attractive force. From Eqs. (9) and (10) we obtain an equation for the Newtonian potential Then we can see that the scalar field adds an additional attractive force to the Newtonian potential if 3+2 ω BD > 0 and, consequently, µ becomes larger than one. On the other hand, for 3 + 2 ω BD < 0 the scalar field mediates a repulsive force. This is related to the fact that for 3 + 2 ω BD < 0 the scalar field becomes a ghost that has a kinetic term with the wrong sign. In this case, the scalar field suppresses the Newtonian force and µ be- comes smaller than one. Next, let us discuss the effect of the mass. For 3 + 2 ω BD > 0 the solution for the scalar field in case of a spherically symmetric source becomes ϕ ( r ) = − 1 3 + 2 ω BD r$_{g}$ r exp [ − r r$_{M}$ ] , (19) r$_{M}$ ≡ √ 3 + 2 ω BD M 2 (20) where is the Compton wavelength of the scalar field. On large scales, r > r$_{M}$ , the solution exponentially decays and the scalar field becomes negligible. In this case there is There are two ways to recover GR. One is to take | ω BD | ≫ 1 and the other one is to consider large scales Ma/k ≫ 1. Q ( k, a ) = 2(1 + ω BD ) + M $^{2}$a$^{2}$/k2 3 + 2 ω BD + M $^{2}$a$^{2}$/k2 , (14) where k$_{ij}$ = k$_{i}$ + k$_{j}$ . Non-linear terms become important when considering non-linear clustering of dark matter and they are crucial to recover GR on non-linear scales. However, in this work we are interested in linear pertur- bations thus we only take into account the linear term. From Eqs. (9)-(13) we can directly derive the functions Q ( a, k ) and η ( a, k ): η ( k, a ) = 2(1 + ω BD ) + M $^{2}$a$^{2}$/k2 2(2 + ω BD ) + M $^{2}$a$^{2}$/k2 . (15) I ( ϕ ) = M $^{2}$ϕ ( k ) + 1 2 ∫ d $^{3}$k$_{1}$ d $^{3}$k$_{2}$ (2 π ) 3 δ$_{D}$ ( k − k$_{12}$ ) × × M 2 ( k$_{1}$ , k$_{2}$ ) ϕ ( k$_{1}$ ) ϕ ( k$_{2}$ ) + . . . (13) in a Fourier space. Here the interaction term I can be expanded as (3 + 2 ω BD ) k $^{2}$ϕ = − 8 πG$_{N}$ a $^{2}$ρ$_{m}$ δ$_{m}$ − I ( ϕ ) , (12) In general, modified gravity models that explain the late-time acceleration predict ω BD ∼ O (1) on sub-horizon scales today. This would contradict the solar system con- straints which require ω BD > 40000. However, this con- straint can be applied only if the BD scalar has no po- tential and no self-interactions. Thus, in order to avoid this constraint, the BD scalar should acquire some inter- action terms on small scales. Generally, we expect that the BD scalar field equation is given by [36] Φ + Ψ = ϕ, (9) (3 + 2 ω BD ) k $^{2}$ϕ = − 8 πG$_{N}$ a $^{2}$ρ$_{m}$ δ$_{m}$, (11) k $^{2}$Φ = 4 πG$_{N}$ a $^{2}$ρ$_{m}$ δ$_{m}$ + k 2 2 ϕ, (10)