The inverse square law of Newtonian gravity implicitly assumes a null mass for the graviton and an infinite range of gravitational force, which contradicts the finiteness of the observable universe and the existence of gravitational waves. If local inertia is somehow linked to the large-scale distribution of matter in the Universe (Mach`s Principle), then the inverse square law is insufficient to describe it. Also it´s cannot explain the rotation curves of galaxies, neither the missing mass of Zwichy, and is not enough for to explain the observed baryonic matter density in cosmological scales. So in LFRW cosmological models to appear the paradigmatically concept about non baryonic Dark Matter (cold dark matter). 

By other hand, using the Boltzmann distribution, the calculus of the mean energy per nucleon during the surface of last scattering, leads to  at  complementary contribution to the Newton gravity. This inverse Yukawa-like field (UYF) would be caused by distribution at great scale of the baryonic mass, in the sense of Mach's principle. This UYF-field result: null in very near solar system, slightly attractiveness in ranges of interstellar distances, very attractiveness in distance ranges comparable to galaxies cluster and repulsive to cosmic scales

The consequences of adding a UYF-field as a large-scale contribution of Newtonian gravity [5] implies that:

1. The cold dark matter could be interpreted as the gravitational contribution caused by the large scale distribution of the ordinary baryon matter:

2 The dark energy would be caused by the cosmic acceleration in local frameworks, by the large scale distribution of the ordinary baryon matter, as prescribed the Mach's Principle

3 The theoretical upper limit for Hubble constant leads , then the age of the universe results in order of the   ~ 11.42 Gyr, with very-good agreement with the White Dwarfs Cosmo-chronology.

4 The Virial theorem and Kepler’s third law now including an additional term (FYF) that solve the Zwicky’s paradox, so  the “missing mass” is reinterpreted as the  energy  of the UYF-field,  and rotation curves of galaxies problem could be modeled similarly to the Milgrom models.

5  In the approach of the UYF-field  the gravitational component of the redshift  increase the gravitational redshift until an factor of ~8,  it´s solvent the Arp’s controversy,  and would help understand the problem of AGN at High-redshift and the nature of Quasars.

6 The inclusion of the UYF-field leads to a reduction in the calculation of the angle of deflection of the gravitational lensing so the deflecting mass at long distances would have been underestimated by a factor of three, without affect the deflection in total eclipses of sun.

7 The UYF reduces the angular diameter distance for all z-values with a maximum in z≈2, in angular diameter distance distribution.

8 In range of comoving distance of the megaparsec, the Jeans Mass decreases then a perturbation will collapse even further when compressed, leading to run-away collapse of the protogalaxies..

9 The large-scale modification of Newtonian gravitation given an identical result for BAO as the paradigm of cold dark matter, by the arithmetic identity:  Ωm=Ωb+Ωc= Ωb(1+ΩYF); but it does not require assuming the existence of non-baryonic exotic particles

10 The CMB and primordial nucleosynthesis remain unchanged by a large-scale modification of Newtonian gravity, such as the UYF-field discussed here.

Regardless of whether the expression proposed for the so-called Yukawa-type inverse field (UYF) is exactly the one suggested here, it has been demonstrated that including an additional gravitational term—varying with comoving distance—would allow for the explicit incorporation of Mach's principle as a theoretical alternative to the non baryonic dark matter paradigm. This large-scale modification of gravity is compatible with FRW cosmology and astronomical observations, while also predicting a finite range for gravity.