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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] 

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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 

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where the function 

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 / $^{3}$. 

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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 ) ): 2 / 3 

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where 

An example of the relation (17) for several α ’s and r$_{c}$ is shown in Fig. 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]). 

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