3 If the Vainstein radius r$_{V}$ ≪ R$_{ta}$ for halos of all masses that are experimentally observed (roughly from ∼ 10 $^{8}$M ⊙ to ∼ 10 $^{16}$M $_{⊙}$), then for distances r ≫ r$_{V}$ the corrections to the Newtonian potential reduce either to the order one renormalization of the gravitational constant (on the “nor mal branch”) or become indistinguishable from the Λ -term (“self- accelerated branch”). In both cases S ( M ) ∝ M 1 / 3 with the normalization, different from the pure Newtonian case. In the opposite case r$_{V}$ ≫ R$_{ta}$ , one can utilize the per- turbative expansion of the function π ( x ) . The gravitational potential well inside the Vainstein radius is given by [3, 4] where c$_{1}$ ∼ O (1) and is positive for the “normal branch” and negative for the “self-accelerating branch” [5] in full ana logy with the DGP model. Notice that a > 1 for α < 3 $_{4}$. Using expansion (12) one arrives to where the function I$_{1}$ ( α ) is shown on the Fig. 2. Substituting the expression (13) back into equation (9) and using (7), we obtain As x$_{ta}$ ≪ 1 and a > 1 , one finds that where to compute x$_{ta}$ we use Eq. (7) with ρ$_{0}$ instead of ρ$_{ta}$ . From Eqs. (7) and (15) we see once again that for all α = 1 2 (i.e. β = 2 ) the turn-around density ρ$_{ta}$ loses its univer- sality and becomes the function of the halo mass M . The turn-around radius R$_{ta}$ ( M ) is related to ρ$_{ta}$ via R$_{ta}$ ( M ) = ( M/ρ$_{ta}$ ) 1 / $^{3}$. Clearly, the most interesting case is when r$_{V}$ ≈ R$_{ta}$ for some range of observed halo masses. In this regime the de- viations from Newtonian gravity become the strongest. The range of values r$_{c}$ for which this happens is shown in Fig. 4 (the value of t$_{0}$ is chosen to be the lifetime of the Universe t$_{0}$ ≃ 1 . 3 × 10 10 years). We expect that for r$_{c}$ in the region Fig.4 the slope of the relation S ∝ M κ will change. Anal- ysis of this case requires however an exact solution of the non-linear analog of the Poisson equation in theories with α (see e.g. [4]), i.e. the knowledge of properties of the func- tion π ( r/r$_{V}$ ) in the range of radii, where the perturbative expansion (12) breaks down. Notice, that the region where r$_{V}$ ≈ R$_{ta}$ shrinks toward the value r$_{c}$ = 2 $_{π}$t$_{0}$ as α → 1 $_{2}$. For this value of r$_{c}$ the Vainstein radius in the DGP model is equal to the turn-around radius for all halo masses . 1 Knowledge of the potential π ( x ) would also allow to probe the modifications of gravity directly in the Local Group and other nearby galax- ies, by studying the infall trajectory around the turn-around radius (see e.g. [29]). An example of the relation (17) for several α ’s and r$_{c}$ is shown in Fig. 3. Under the assumption of exact self-similarity, discussed above (i.e. r$_{C}$/R$_{ta}$ = const ) one arrives to the following expression for S (recall that β $^{= 4(1}$− α ) ): 1 In general for r$_{c}$ ∼ H − 1 0 the turn-around radius is smaller than r$_{V}$ for α < 1 2 and bigger than r$_{V}$ for α > 1 $_{2}$. FIG. 3: Examples of S − M relation in theories with α < 1 / 2 together with the data from [14]. FIG. 2: Function I 1 ( α ) defined in Eq. (13) 0.2 0.4 0.6 0.8 1.0 Α GLYPH<Minus> 0.5 0.5 1.0 I$_{1}$ GLYPH<LParen1> Α GLYPH<RParen1>  0  1  2  3  4  5  6  7  8 10 7 10 8 10 9 10 10 11 10 12 10 13 10 14 10 15 10 16 DM column density, log 10 [S/M sun  pc -2$^{]}$ DM halo mass [M$_{sun}$] Clusters of galaxies Groups of galaxies Spiral galaxies Elliptical galaxies dSphs Norm. branch, = 0 r$_{c}$ 150 Mpc Self-acc. branch, 0 r$_{c}$ 150 Mpc Norm. branch, = 1/4 ; r$_{c}$ 300 Mpc Self-acc. branch, 1/4 ; r$_{c}$ 300 Mpc Best-fit model S M 0.23 π ( x ≪ 1) ≈ 1 + c$_{1}$x a ; a = β + 1 2 = $^{5}$− 4 α 2 (12) I ( x$_{ta}$ ≪ 1) ≈ 1 + c$_{1}$ 2 x a ta   $^{2}$ π ∫ 1 0 dx $^{1}$− x a − 1 ( 1 x − 1 ) 3 / 2    ︸ ︷︷ ︸ ≡ I$_{1}$ ( α ) (13) ( ρ$_{0}$ ρ$_{ta}$ ) 1 / 2 ( 1 + c$_{1}$ 2 x a $_{ta}$I$_{1}$ ( α ) = 1 (14) ρ$_{ta}$ ≃ ρ$_{0}$ ( 1 + c$_{1}$I$_{1}$ ( α ) x a $_{ta}$( ρ$_{0}$ ) (15) where M$_{lim}$ ≡ 1 G [ ( r$_{c}$ 2 ) 3 β ( π t$_{0}$ ) 2(1+ β ) ] 1 β$_{−}$ 2 (18) S ( M ) = ρ$_{C}$r$_{C}$ ∝ M 1 / $^{3}$ρ 2 / 3 ta (16) ∝ M 1 / $^{3}$ρ 2 / 3 0 ( 1 + 2 3 c$_{1}$I$_{1}$ ( α ) ( M M$_{lim}$ ) 1 − 2 α 3 ) (17)