The Mach's Principle and large-scale modification of Newtonian gravitation as alternative approach to Cold Dark matter and Dark Energy

: We take a heterodox approach to the ΛFRW Cosmology starting from the modification of Newtonian gravity by explicitly incorporating Mach's Principle through an additional term a great scale in the gravitation. The results show that at the after of the matter-radiation decoupling, the distribution of matter at scales greater than 10Mpc contributes with an inverse Yukawa-like field, which verifies the observations: resulting null in the inner solar system, weakly attractive in ranges of interstellar comoving distances, very attractive in comoving distance ranges comparable to the clusters of galaxies, and repulsive in cosmic scales. This additional term explains dark energy, re-moves the incompatibility between the density of matter and the flatness of the universe; and also allows the theoretical deduction of the Hubble-Lemaitre Law. Additionally, Birkhoff Theorem, Virial Theorem, the missing mass of Zwicky, the BAO, gravitational redshift are discussed. It is concluded that the dark energy and the missing mass can be approached with the usual physics if a classical, large-scale modification of the Inverse Square Law.


Mach Principle and Phenomenology
In the book "The Science of Mechanics", Ernst Mach [1] concludes that the local inertial frame is determined in some way by the movement of distant astronomical objects. This conclusion, known as Mach's Principle, arises from the very exact coincidence of the two ways of measuring the rotational speed of the angular velocity of the Earth: dynamically (Foucault's Pendulum) and astronomically (with respect to the "Fixed stars"), as indicated in the figure 1. This coincidence, together with Newton's well-known "bucket experiment", according to which the curvature of a rotating bucket with water is an expression of inertia ("vis ínsita" in Newton's words).
Inspired by these empirical facts, E. Mach, postulates those inertial forces owe their existence to the distribution of distant matter in the Universe. In other words, the water in Newton's bucket would not bend if there were no galaxies and it is the teleconnection that exactly equates the period of Foucault's pendulum at the North Pole with the sidereal day. As a result, if Newton's bucket was still on the Earth, and put the Universe in its entirety to rotate, it would also bend the surface of the water. Mach's thought decisively influenced Einstein's thought as he himself refers in his autobiographical notes [2] "It was Ernst Mach who with his History of Mechanics ... exerted a profound influence on me ... also during my young years, Mach's epistemological position influenced me a lot, a position that today seems essentially untenable".
The General Theory of Relativity (TRG) when connecting gravitational fields with inertia, seem to give entry to Mach's Principle, through the Field Equations. But they do not fully verify it, because the limit conditions of the Field Equations. From Einstein's point of view, the "relativity of inertia" was an integral part of his General Theory [3], that is to say: the inertial field defined by the metric tensor (g  ) had to be completely determined by the distribution of masses and energies in the Universe (T  ); but this is not completely true because the relationship established by the Field Equations are differential and it is necessary to impose limit conditions at infinity to unequivocally determine the g  from a given T  . This need to impose additional conditions prompted the incorporation of the cosmological constant in Einstein's Model of the Universe, which we now incorporate through the Friedmann-Robertson-Walker models with cosmological constant (ΛFRW model). If local inertia is somehow linked to the large-scale distribution of matter in the Universe, then the inverse square law of Newtonian gravitation is insufficient to describe it. The gravitational field at any point in our galaxy or at any other point in space (a point in the local group of galaxies for example) would be the sum of the gravitational contributions of all the other galaxies at that point; and although these contributions were insignificant, their total sum is not necessarily is null. So the gravitational interaction between two stars would be the one prescribed by inverse-square Newton's Law plus an additional contribution from the distant masses, which does not have to be the same for different points in space; since, on a large scale, the distribution of masses is not spherical with its center in the local frame of observation.
By other hand, assuming that the dynamics of the universe is prescribed only by the Newton law of gravity we encounter serious difficulties in describing the dynamic of the Universe [4]: (i) It cannot explain the rotation curves of galaxies, which show its incompatibility with the virialized masses (ii) the mass observed into the rich clusters of galaxies is significantly less than that required to maintain these systems gravitationally stable, and (iii) In cosmological scales, the observed baryonic matter density is much lower than predicted by the FRW models with cosmological constant. The problem of missing mass appears to affect the dynamics at all length scale beyond the Solar System. One solution has been to propose the requirement of missing matter of unknown origin (non baryonic Dark Matter) with equally unknown properties and only interacts gravitationally with ordinary matter. However, after more than a two decades of strenuous efforts: theoretical, astronomical observations and laboratory experiments, only their existence has been suggested conjectural or paradigmatically.
It is worth asking, then, if any modification of the Law of Gravitation is possible that on a large scale solves the dynamics observed in the universe (including cosmic acceleration: Dark energy), and also, agrees with the certainties of Newtonian gravitation. Such a modification of gravitation should agree, at the scale of the solar system, with the inverse-square law and be compatible with terrestrial experiments (Eövos-like experiment).
The purpose of this work is to deduce a possible modification of the Law of Gravitation that meets these requirements, that is, that all particles with nonzero rest mass, is subject gravitational inverse-square law, plus an additional term that varies with the comoving distance. This complementary contribution to the inverse-square law, would be caused by distribution at great scale of the baryonic mass, in the sense of Mach's principle. This inverse Yukawa-like Potential [4,5], 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 .
For this, the mean energy per nucleon during the decoupling between material and radiation in the early universe is discussed (section II) and the form of the Yukawa-like potential is deduced together with the coupling constants, consistent with the previously reported parameters [5,6]. The implications of Birkhoff's theorem and the general form of the potential are shown in section III. The cosmological and astrophysical implications are summarized in section IV, together a preliminary study of its implications in the Baryonic Acoustic Oscillations (BAO) and the Cosmic Microwave Background (CMB), before the conclusions in the last section.

Matter-radiation decoupling
During recombination, in the first moments of the formation of the Universe, when the material is separated from radiation and hydrogen is synthesized (surface of last scattering), the average energy per unit mass (U) can be expressed for each nucleon a the temperature T of the plasma, using the Boltzmann distribution, as the work required to move the proton from the initial position to the comoving distance r relative to the center of the protogalactic cloud of mass M , as : Where denotes the Boltzmann constant, U is an constant and is the proton energy. This energy is only kinetic energy, which could be expressed in terms of the gravitational energy of the protogalactic clouds that forms later, thus Where, mp is the rest-mass of proton. Notice that we can use Virial theorem in its usual form, because we are at z = 1100 and the first protostars and protogalaxies that appear much later, around z = 6, have not yet formed.
Using (1) and (2), then Where is the average plasma temperature at the moment of the decoupling ( ≈ 13,6 ) . M and is the mass and average radius of protogalaxy respectively, and is the critical density. ≡ ( ) = 4 GM r is a coupling constant in units of J/kg (l≡ 1 m -1 is a dimensional parameter) [5,6] because it is the gravitational potential in upper limit when ε/T→0 in the protogalaxy of mass M.
By other hand, the energy balance in the protogalaxy demands that = Because the contribution of the large scale of gravitation (UYF) is null in ≈ . Remember that U=UN+UYF, i.e. the total gravitational energy is the Newton gravity plus the large scale contribution (UYF-field).
Using the critical density [4][5][6]    (3) with r in Mpc, as: Also, the force per unit mass (acceleration), complement to large-scale of the Newtonian gravitation is : as the previous report [5,6], according to the assumption to the finite range of the gravitation; i.e. with massive graviton [5]: And the maximum rm occurs in This value is in order of the Abell radius, for the typical clusters of galaxies; as the previous assumption [5]. We can see in (7), that if r is negligibly small compared to r=2.5 Mpc, then FIY≈0, i.e. the gravity interaction is only prescribed by the inverse square law of gravitation, in accordance with Eövos-like experiments.
For ranges the comoving distances, between objects gravitationally bound, with r smaller we obtain FIY ~ r -1 , see for details [5], therefore, the MoND-Milgrom results are recovered as a particular case, and could be applied to solve the problem of the rotation curves of galaxies.

Birkhoff's theorem and Overview
The general solution to the gravitation Poisson equation, under spherical symmetry, depends on the mass distribution outside of r: In the Newtonian gravity approach it is easily understood that the second term in (11) is canceled because of the fact that the solid angles extending from one point within a sphere to opposite directions have areas in the sphere that escalade as r 2 , while The gravitational force per unit the dough scales such as r -2 , so that the gravitational forces of the two opposing areas are canceled exactly. The equivalent in general relativity is called Birkhoff's theorem. But in general approach, if the gravitation field has a large scale contribution, then: Thus, it`s no true that the second term in (11) 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 (12), i.e. the UYF given in (3), 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 [7] 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 [8], 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 [9]. 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 ().
As example, in figure 3 is plotted the effective gravitational energy, per unit mass, for various members of the local group of Galaxies, (Equation 12). In the left panel, for very close satellite gal-axies (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.

Cosmological and astrophysics consequences
We consider a model of the expanding universe, where each galaxy is a particle of a perfect fluid, under the hypotheses of homogeneity and isotropy; i.e. a FRW cosmological model. The cosmological term leads to usual Friedmann equations [4,5]: In this approach the Dark Energy can be thought of as a "cosmic acceleration" ( the cosmological constant as an force by unit of mass): We obtain [5,6] in flat universe model (k=0): where it has been used the standards cosmological parameters: Ω = ⁄ , Ω Λ = Λ 2 3 ⁄ , q 0 = −̈⁄ for the density of matter, cosmological term and deceleration respectively; and the definition Ω IY ≡ Λ( ) 2

About the Dark Energy
When rrc, the comic parameter Λ( ) in the intergalactic scale, is using (15): Thus, in the present model, the dark energy would be the cosmic acceleration in local frameworks, caused by the large scale distribution of the ordinary baryonic matter, as prescribed the Mach's Principle, through of UYF proposed. As before, replacing (20) in cosmological density parameter then (see [5] for detail): Using here the upper limit for Hubble parameter h=0.863 [6], we obtain ΩΛ ≈0.72 in good agreement with the measurements of type Ia Supernovae.

The Hubble-Lemaître's Law and the Age Universe
Consider the photons emitted from a remote galaxy with recession velocity v, and their observation in the reference local frame. Therefore, we should evaluate (3) at r  50Mpc, with initial condition v = 0 in t = 0. We find: Also, the Friedmann equation can be written as: Note that it is the ΛFRW conventional equation for the age of universe, because of

The missing mass of Zwicky
Let us consider the Clausius's Virial expression = ∑ ⃗ ⃗ and deriving with respect to time, and then averaging with respect to a complete period (τ), we obtain the well-known virialized expression between kinetic energies and power. But now, the gravitational potential energy is given by (12) then [5] : An important result is that (25) solve the Zwicky's paradox. The "missing mass" could be interpreted as the energy associated with the UYF field. For example in the particular case of the Coma cluster then:

Gravitational Lensing
Let us consider the deflection of a point-like lens of mass M, under the assumption of basic thin gravitational lent, when gravitational potential is small, the effect of the space-time curvature on light trajectory can be described as an effective refraction index η, given by: The deflection angle (ζ) of the light rays which traveling in a gravitational field is given by the integration of the gradient component of η orthogonal to the trajectory: But now, the gravitational field  is given (12), assuming that is small, i.e. /r<<1 we obtain: Where, we used 0 / r0, and ( ) = 4 as before.
If b is the impact parameter of the unperturbed light ray and denote the position along the unperturbed path as measured from the point of minimal distance from the lent: = + , Where we using previous notation, RP denote the physical radius of stars and Rs is the Schwarzschild radius, we find the deflection angle: Note that the inclusion of the potential (12) leads to a reduction of one third in the calculation of the angle of deflection of the gravitational Lensing. Consequently, estimates of the deflecting mass in gravitational lenses observed at long distances (where the large-scale correction for Newtonian gravitation makes sense) would have been underestimated by a factor of three. Obviously, these do not affect the observation of the deflection of light, in the case of a total solar eclipse, because the UYF is null at the scale of the solar system (figure 2) because comoving range of distance is much smaller than kiloparsecs.

Longitude and Mass of Jean
A self-gravitating, proto-stellar or proto-galactic mass, remains in hydrodynamic equilibrium as long as its dimensions do not exceed the length of Jeans. The Jean's length (J 0 ) is the characteristic length for which pressure balances gravity; which in term of the density (), temperature (Te) and hydrogen mass (mH) is: Consider a uniform self-gravitating mass, with electronic temperature and effective density, constants. In the Newtonian Gravitation approach, the Jeans length ( ) remains constant for all r and therefore agglutination and / or fragmentation will be homogeneous. Figure 5 shows that, when considering a large-scale modification of Newtonian gravitation, through the potential UYF, the effective Jeans length (λJ) decreases for the mega parsec range, reaching a minimum of approximately half the Abell radius favoring the agglutination.
The reason for this is that the gravitational potential now exerts a stress on the gas, between the center (where the term Newtonian r-2 dominates) and the periphery (where the term of the force due to UYF dominates) whose maximum is in order of Abell: 1.2 Mpc.
As the Jeans Masses, is proportional to the cube of Jeans longitude, would be decrease very much. The Jeans mass is the critical mass above which gravity dominates, so if the Jeans Mass decreases then the gravity dominates, and so a perturbation will collapse even further when compressed, leading to run-away collapse in range of comoving distance of the megaparsec.

Angular Diameter distance
The angular diameter distance is, using (24) and the redshift dependence of universe-scale factor: In Fig. 6, we shown the dimensionless angular diameter with and without UYF (dashed lines), and only the parameter of the matter density (continues line: Λ=0) . We can see that the UYF reduces the angular diameter distance for all z-values and their maximum in z≈2. Figure 6. Angular diameter distance in function of the redshift [5] 4.9 BAO and CMB anisotropies The early universe consisted of hot plasma of photons, electrons and baryons closely coupled by Thomson scattering, with oscillations in the photon fluid, due to radiation pressure and gravity. The essential physics of these linear perturbations can be study from non-relativistic hydrodynamic approach, through equations of continuity, Euler`s and Poisson`s equations, in Lagrangian form. Now consider a small perturbation (first order) superimposed on the "background" fluid, then the fractional density perturbation (or density contrast) δ/0 obeys the relations [16]: In the present UYF -field formalist the dependence of density and pressure remain unchanged on the scale factor of expansion R(t), and therefore, not affects the Early Universe. Thus, during the decoupling matter-radiation, the primordial fluctuations and the anisotropies in the CMB, the nucleosynthesis (baryogenesis), should remain unchanged by a large-scale modification of Newtonian gravity. Remark that UYF is forty orders of magnitude higher than the average distance per nucleon in the primordial plasma. Also the Sachs-Wolfe effect also does not change, because the size of the horizon at the time of recombination is approximately 100 kpc, much less than the maximum range of the gravitational force with massive graviton (~10 Mpc) and at such ranges, the graviton would travel the entire universe inside the horizon without decay.

Summaries and Conclusion
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 FRW 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 baryonic matter: ( ) The dark energy would be caused by the cosmic acceleration in local frameworks, by the large scale distribution of the ordinary baryonic 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 UYF-Field is exactly the one proposed here or not, is that, perhaps, to understand the dynamics of the universe, we would have to abandon the assumption about the universal validity of the inverse square law of Newtonian gravity, assumed uncritically, instead of conjecturing unobservable exotic matter Funding: "This research received no external funding".