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We deduced the Hubble law and the age of the Universe, through the introduction of the Inverse Yukawa Field (IYF), as a non-local additive complement of the Newtonian gravitation (Modified Newtonian Dynamics). As a result, we connected the dynamics of astronomical objects at great scale with the Friedmann-Robertson-Walker ΛFRW) model. From the corresponding formalism, the Hubble law can be expressed as
*v *= (4
π[G]/c)
*r*
, which was derived by evaluating the IYF force
at distances much greater than 50 Mpc, giving a maximum value for the expansion rate of the universe of *H*_{0}
^{(max) }≈ 86.31 km·s^{-1}Mpc^{-1}, consistent with the observational data of 392 astronomical objects from NASA/IPAC Extragalactic Database (NED). This additional field (IYF) provides a simple interpretation of dark energy as the action of baryonic matter at large scales. Additionally, we calculated the age of the universe as 11 Gyr, in agreement with recent measurements of the age of the white dwarfs in the solar neighborhood.

The idea of a model for a universe in continuous and constant expansion emerged from the pioneering work of Hubble, Slipher and Humason [

Hubble law was empirically proposed by Hubble [

where

Since the discovery of the accelerating expansion of the universe through the study of high red shift supernovae [

Although Hubble law represents the first observational test of the expansion of the universe, and today supports the actual cosmological model [

One of the biggest problems in the Big Bang cosmology, closely linked to the expansion of the Universe and the Hubble law, is the evidence of the accelerated expansion of the Universe, commonly referred as dark energy, whose understanding is still far from complete. Also, the inconsistency between the observed average density of matter and the density required for flatness of the universe, a problem known as the missing mass, has become the paradigm of the hypothetical non-baryonic dark matter. This discrepancy between the astronomical observations of the density of matter and expected in ΛFRW model in the Big Bang theory, has prevailed in the last years. An alternative to the paradigm of non-baryonic dark matter is the theory of Modified Newtonian Dynamics (MoND), which involve changes in the Newton’s law of gravitation (inverse square law).

In this sense, one possibility to solve both problems: dark matter and dark energy, is the non-local gravitation recently proposed by Falcón [

The inclusion of this second term in the force of gravity, consistent with Eötvös-like experiments, can reconcile the ΛFRW model with observables of the Big Bang, without the paradigm of non-baryonic dark matter. Additionally, gives an explanation for dark energy, and allows us to theoretically deduce the Hubble law.

In this paper, we will show that the Hubble law can be derived from the MoND theory proposed by Falcón in a natural way through the corresponding condition of cosmological scales. To this end, in Section 2 we will review the paper of Falcón emphasizing the repulsive behavior of the non-local gravitational field at large scales, giving a starting point for deducting the Hubble law. The theoretical deduction of the Hubble law and even an analytical determination of the Hubble constant will be given in Section 3. In Section 4, we will contrast the determined Hubble constant with the observational data of 392 objects selected from the NASA/IPAC Extragalactic Database (NED), also a brief discussion about the cosmic age problem is given. Finally, the conclusions are given in Section 5.

Current Big Bang cosmology assumes Newtonian gravitation as the only fundamental force at astronomical scales, giving a complete determination of the dynamics of the universe. However, from this idea we face serious difficulties to describe the behavior of the Universe: 1) galaxy rotation curves are not explained without the inclusion of non-baryonic dark matter, whose fundamental nature and properties are completely unknown; 2) into the rich galaxy clusters, the observed mass of stars and the gas mass inferred from the X-ray diffuse emission is significantly less than that required to hold these systems gravitationally stable; and 3) the accelerated expansion of the universe violates our understanding about how gravity works at cosmological scales (see [

The simplest way for modeling the accelerated cosmic expansion is by introducing a cosmological constant into the Einstein’s field equations so it can represent a hypothetical negative pressure of the vacuum of space, also called dark energy. However this is given as a disconnected idea from the dynamics of the astronomical objects, which is limited to the Newton’s law of gravitation.

While Newtonian gravitation (inverse square law) has been highly supported by laboratory experiments and satellites, there is no experimental evidence to confirm its validity beyond the Solar System [

where

Then, the proposed modification considers the contribution of both the Newtonian and the non-local gravitational field, so that the dynamics at all scales is determined by the force per unit of mass as

where it is important to note that there is a dependence on the baryonic matter only.

In particular, a zero contribution of the non-local term can be verified at distances below

solving the galaxy rotation problem. Also, the non-local IYF, evaluated in the Abell radius

From

providing an asymptotic cosmic acceleration, consistent with the observations. This opens the possibility to describe the behavior of the cosmological constant by setting it as a dynamical term with the form

where the dot denotes the time derivate of the scale factor

The introduction of the non-zero contribution of the cosmological constant brings a modification to the usual form of the matter density parameter,

where the dark energy density parameter (or cosmological density parameter),

For a complete interpretation of the behavior of the IYF potential and details about the cosmological consequences by adding the IYF to Newtonian dynamics and to FRW cosmology see [

A numerical value for

Usually, the Poisson equation is written for the Newtonian gravitational case as

with

and the same for the matter density

taking into account that this equality works for

Here, we note that the obtained magnitude,

On the other hand, in Section 2 we saw that the IYF, as anon-local term, shows a repulsive behavior at cosmological scales

Consider a particle (galaxy, galaxy cluster, nebulae, etc.) with nonzero rest mass under the influence of the IYF force. The contribution of the Newtonian gravitational force is not important at cosmological scales (i.e. at

where

Then, it is possible to obtain an expression of the velocity by integrating Equation (12) as follow

of the particle. Therefore,

With

where without loss of generality we assumed the initial condition

so that the Hubble law can be written as

Note that Equation (16) basically is equal to Equation (1), establishing a linear relation between recessional velocities and distances for a given particle (galaxy, cluster of galaxy, nebulae, etc.), just under the assumption of cosmological scales, in agreement with the current cosmological model. Additionally, the limit value of the linearly constant gives the maximum expansion rate of the universe, again as product ofstudy distances much greater than

In the next section, we will test the

Although the first determination of the Hubble constant was

In this section, we will contrast our value for the Hubble constant, of

In order to verify the Hubble law and our value for the Hubble constant, we will use the primitive technique used by Hubble, which consists in plotting the observational measurements of the velocity (via red shift) and the distance of a set of objects such as galaxies, quasars, radio sources, X-ray sources, infrared sources, etc. For this end, we considered the Master List of Redshift-Independent Extragalactic Distances of 15339 galaxies provided by NED (Version 9.2.0). The observational measurements were filtered by: 1) recent measurements (year of publication from 2009); 2) distantobjects

A Hubble diagram for the 392 galaxies, in a range of 50 - 1400 Mpc, is shown in

In

On the other hand, an additional result can be obtained through the determined Hubble constant: the age of the universe,

[

when the IYF is zero.

Numerical integration of Equation (17), for

The inclusion of a long-range component in the law of gravitation allows linking the Hubble law with the dynamics of the large-scale Universe. Particularly, if the non-locality of gravitation is included through apotential as shown here, Yukawa Inverse type, we can connect the dark energy with cosmological constant and derive from there the Hubble law, consistent with the formalism of the Big Bang, and astronomical observations, without resorting to the paradigm of non-baryonic dark matter, or an “exoticphysics”.

The inclusion of a long-range component in the law of gravitation, through an inverse potential Yukawa-like, represents the collective contribution of the gravitational effects of large-scale, on the order of tens of megaparsec caused by ordinary baryonic matter. In this sense, the IYF explicitly includes the Mach principle in the formalism of FRW cosmology, as Einstein pretended with the Theory of General Relativity.

The prescription of the Hubble constant in terms of the fundamental constants, as in Equation (15), appears to correspond to the observational data for distant objects, whose distance and red shift are independently known; as we can see in

For the nearest objects, with distances less than a hundred megaparsec, the Hubble constant would seem less than true value, because in these ranges, the contribution of the IYF field is less, as was shown in

A current Hubble constant

This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.