If the scale r$_{c}$ is of the order of ∼ H − 1 0 , such modifications of gravity can provide an explanation for the late-time cosmo- logical expansion of the Universe [1, 2]. The corrections to Newton’s law become negligible as r → 0 ( π (0) = 1 ) and the radius (6) characterizes the scale where the deviations from Newton’s potential become of order unity. Using the relation (where ρ = M/r $^{3}$) we find that among the theories of mod- ified gravity (5) only β = 2 ( α = 1 $_{2}$, the DGP model [1]) possess the property (4) and consequently (2). The property (2) can be probed experimentally. Exten- sive catalog of DM-dominated objects of all scales, collected in [14], exhibits a simple scaling relation of the properties of the DM halos. Dark matter distribution in the majority of ob- served objects can be described by one of the universal DM profiles (such as e.g. NFW [21] or Burkert [22]). Such pro- files may be parametrized by two numbers, directly related to observations – a characteristic radius r$_{C}$ (off-center dis- tance where the rotation curve becomes approximately flat, equal e.g. to r$_{s}$ for NFW) and a DM central mass density ¯ ρ$_{C}$ , averaged inside a ball with the size r$_{C}$ . It was shown that DM column density S ∝ ¯ ρ$_{C}$r$_{C}$ , (see [13, 14] for a de- tailed definition) changes with the mass as S ∝ M $^{κ}$, where κ ≈ 0 . 22 − 0 . 33 . It was demonstrated in [13] that in the simplest self-similar model (i.e. assuming that r$_{C}$/R$_{ta}$ is the same for the DM halos of all masses) property (1–2) implies a scaling S ∝ M 1 / $^{3}$, compatible with observations. Observa- tions (see e.g. [23–27]) demonstrate that the ratio of r$_{C}$ to the virial radius depends weakly (as M ≈− 0 . $^{1}$) on the mass of DM halos. The Fig. 1 shows ratio of the virial radius of a halo to its r$_{C}$ for DM density profiles from the catalog of [14]. These results are in perfect agreement with the Λ CDM numerical simulations (see e.g. [18, 28]). Due to this slight deviation from the self-similarity, the best fit value of the scaling pa - rameter κ = 0 . 23 (see [13] for discussion). For the qualitative discussion of this work, it is important that S ( M ) is a fea- tureless power-law dependence, whose slope does not depend on mass and that the deviation from the slope 1 3 is small (as follows from observations). We can conclude that in the theories, satisfying the con- dition (5) (e.g. in DGP model) the properties of DM halos theory will follow the same scaling relation S ∝ M 1 / 3 and the difference with the Λ CDM case will only be in a different normalization of this scaling relation. For other theories, de- scribed in [3, 4], with α = 1 $_{2}$, we can see from the Eq. (7)) that the potential φ$_{α}$ ( r ) is not of the form (4) and we can expect deviation from the universal scaling law. Let us work out the S − M for a general α in details. A general expression, relating the turn-around time, turn-around radius and mass within this radius follows from energy con- Further analysis of Eq. (9) depends on the form of the func- tion π ( x ) . Its exact form is not known (apart from the DGP case). Let us start with analyzing several limiting cases. The derivation of Eq. (9) demonstrates that for all theories of gravity of the form (4) (including Λ -term and the DGP model) ρ$_{ta}$ is a function of ρ$_{0}$ only and does not depend on the mass/size of a particular halo. As a result S ∝ M 1 / 3 (see [13] for details). Here the constant ρ$_{0}$ is a function of lifetime of the Universe only and does not depend on parameters of a dark matter halo. This gives a desired relation between a turn-around density and the life-time of the Universe in the pure Newtonian cos- mology (without cosmological constant). The function I ( x ) is defined in such a way that in the Newtonian limit π ( x ) = 1 one gets I ( x ) = 1 . The solution of this equation gives us the “density” ρ$_{ta}$ ≡ M/R 3 ta as a function of M . When r$_{c}$ → ∞ the turn-around density ρ$_{ta}$ becomes where dimensionless ratio x$_{ta}$ ≡ R$_{ta}$/r$_{V}$ and the function I ( x$_{ta}$ ) is given by Using the general form (5) we can rewrite the expression (8) in the following form: servation and is given by FIG. 1: Comparison of c vir = R vir /r C as a function of DM halo mass for observed DM density profiles from the catalog of [14] . 2 r r$_{V}$ = ( r 1+ β 2 GMr β c ) 1 1+ β ∝ M β$_{−}$ 2 3(1+ β ) 1 ρ 1 / 3 ( 2 Gr β c ) 1 1+ β (7) t$_{0}$ = 1 √$_{2}$ ∫ R$_{ta}$ 0 dr √ φ$_{α}$ ( r $^{)}$− φ$_{α}$ ( R$_{ta}$ ) (8) t$_{0}$ = ( π $^{2}$R3 ta 8 GM ) 1 / 2 I ( x$_{ta}$ ) (9) I ( x$_{ta}$ ) = 2 π ∫ 1 0 dx ( π ( xx$_{ta}$ ) x − π ( x$_{ta}$ ) 1 / 2 (10) ρ$_{ta}$ = M R 3 ta = π 2 8 Gt 2 0 ≡ ρ$_{0}$ (11) I. S − M RELATION FOR GENERAL α  0.1  1  10  100  1e+09  1e+11  1e+13  1e+15 c vir M$_{vir}$ , M$_{Sun}$