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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 

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(where 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- 

## I. S − M RELATION FOR GENERAL α 

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servation and is given by 

Using the general form (5) we can rewrite the expression (8) in the following form: 

where dimensionless ratio x$_{ta}$ ≡ R$_{ta}$/r$_{V}$ and the function I ( x$_{ta}$ ) is given by 

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 

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 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). 

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. 

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