The XXI century cosmology

Inverse Yukawa-like Potential (Falcon 2013, Falcon 2021), would be 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

2. The Model: Inverse Yukawa-like field.

The basic ideas of the cosmological model are raised; starting with the phenomenology and physical arguments (2.1) followed by consistency with Eövos-type experiments and the MoND-Milgrom formalism (2.2). In subsection 2.3 to show a brief derivation of the UYF field during decoupling of radiation from matter in the early universe, then the estimation of the graviton mass (2.4) and ends with a discussion of Birkhoff's theorem and general description (2.5).

2.1 Phenomenology and physical argument.

We assume that any particle with nonzero rest mass is subject gravitational inverse-square law, plus an additional force that varies with distance, caused by Inverse Yukawa-like field (U_YF). Thus the net force of gravitation varies as the law of inverse square in negligibly small distance scales in comparison to the interstellar scale, but it varies in a very different way when the comoving distance is about of the order of kiloparsec or more. In this sense, our argument is a large-scale modification of the Newtonian gravity.The origin of this field (U_YF) is the baryonic mass, like in the Newtonian gravity, and represents the contribution of the gravitational distribution at great scale of the baryonic mass. This potential per unit mass (in units of J/kg) as function of the comoving distance, is heuristically build from a specular reflection of the Yukawa potential (Falcon 2013; Falcon 2011): null in the inner solar system, weakly attractive in ranges of interstellar distances, very attractive in distance ranges comparable to the clusters of galaxies and repulsive to cosmic scales The coupling constants; selected from the astronomical data; are: r_0~50 Mpc (the average distance between clusters of galaxies) and α~2.5 Mpc. We understand that an exact model could will fit the precise values of the coupling constants without modifying the phenomenology. The fig. 1 shows the variation of the U_YF for different ranges of the comoving distance r (in Mpc). This heuristically constructed potential U_YF (r) was constructed phenomenologically to explain the astronomical observables. However in the next subsection 2.3 to shown a simple preliminary derivation of this potential, beginning the mean energy per nucleon during the decoupling between materia and radiation in the early universe, also is deduced the coupling constants, previously reported.

2.2 The Eövos-like experiments and MoND -Milgrom .

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2.3 Matter-radiation decoupling

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Gravitational waves can probe fundamental physics, leading to new constraints on the mass of the graviton. Recent observations of gravitational waves (LIGO and VIGO Collaboration Proyect), have put an upper bound of 1.2×10^−22 eV/c^2 on the graviton's mass

2.4 The mass of graviton

In good agreement with the lower bounds inferred from the observations of binary pulsars, cluster galaxy structure and gravitational lensing (Gazeau & Novello 2011). Also the LIGO Collaboration constrains the Compton wavelength of the graviton as λ_g^0>〖 10〗^13 km (Bergshoeff et al. 2009). The structures found on a large scale in the distribution of galaxies and gas, with characteristic dimensions much greater than 10Mpc; e.g. Sloan Great Wall (Einasto et al. 2011) and Voids (Kovács et al. 2016); do not show symmetric axial distribution that would be expected if gravitation had infinite range. The hot gas in the superclusters of galaxies found by means of the Suyaev-Zel`dovich effect (Planck Collaboration 2011) also does not have a spherical distribution that would be expected if gravitation were of infinite range. In large-scale structures with dimensions greater than 10 Mpc, there is a gravitational bond between the galaxies, by a sequential chain of gravitational attractions between their neighboring components, but not by a common center. Assuming an infinite range for gravity, would imply among other things, to imagine colossal masses for the attractor center in the superclusters of galaxies, which are unobservable (Super Black Hole).

In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. Thus, this part of the theorem is just what we would expect from the fact that general relativity reduces to Newtonian gravitation in the Newtonian limit.

2.5 Birkhoff’s theorem and Overview

Thus, it`s no true that the second term in (14) are null. Therefore, Birkhoff’s theorem could not be applied in the renormalized Newtonian theory of gravitation, such as in the UYF-field. The figure 2 shown the second term of the (15), i.e. the UYF given in (1), for different ranges of comoving distance r, and the phenomenological description. . Note that the potential per unit mass (UYF) results in a smooth function: continuous, differentiable, approaching zero for small range of comoving distance r. The fig. 2 show that UYF has a unique minimum, for the comoving distance in the order of 10 Mpc, as can be expected if the graviton has non-zero rest mass, in according to the detection of gravitational waves (LIGO 2017) i.e. it suggests that the scope of the gravitational force is finite. Newton's law of gravitation implicitly postulates that the range of the force is infinite and consequently, the force would have a greater range than the universe itself, besides, of course, it assumes null rest mass for the graviton. Also, an infinite range of gravity would imply that large-scale structures in the Universe would be spherically symmetric. However, the distribution of hot gas in the superclusters of galaxies, detected via the Suyaev-Zel`dovich effect (Planck Collaborations 2015), and the large-scale structures in the distribution of galaxies, with dimensions greater than 10 Mpc: as e.g. Sloan Great Wall and the Voids, , do not show symmetric axial distribution (Peebles 2020). A finite range of gravity (on the order of 10 Mpc) can explain these structures by a chain sequence of gravitational attractions between their neighboring components and not necessarily by a common center, of colossal masses in the superclusters of galaxies (i.e.: black super holes). It is clear that until today no accretion disks of Super Black Holes , have been detected, despite of advances in X-ray and IR astronomy. The UYF is null in the r-values near the average distance between galaxy clusters (r≈ r0 ~50 Mpc). Besides, this function has only one inflection point; i.e. their derivative (force by mass units) is maximum in r-value in order of the nucleus of Abell's radius (r= rc≈ 1.2 Mpc). And also its derivative (acceleration), is constant at cosmological scales (r>>50 Mpc) such as it would be required to incorporate the cosmological constant (L). As example, in figure 4 is plotted the effective gravitational energy (15), per unit mass, for various members of the local group of Galaxies. In the left panel, for very close satellite galaxies (in logarithmic scale). In the right panel, to show other notable members, of the Local Group, on a linear scale. The galaxy VV124 (UGC4879) maybe the most isolated dwarf galaxy in the periphery of the Local Group, near of the minimum of gravitational energy in accord with present description. Also M3, M33, NGC 300 and NGC55 they have a gravitational potential energy per unit mass, 100 times greater than the spheroid satellite galaxies of the Milky Way. The consequences of this, in dynamic stability and in the description of the rotation curves, is outside the scope of this communication, and would be interesting as an additional test for the approach to the large-scale modification of inverse square law of Newtonian gravity.

Figure 4. Gravitational Energy by mass unit in galaxies of Local Group (Eq.15). (Left) Very near galaxies satellites of Milky Way, on logarithmic scale. (right) Galaxies of the Local group, on linear scale. (Own source).

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