The XXI century cosmology

1.1 The problem of missing mass

(non-baryonic Dark Mass)

The scientific cosmology is based on the description of gravitation by the General Theory of Relativity, more specifically in the solutions of the Friedmannn equations for a model of isotropic and homogeneous universe on a large scale (metric Friedmann-Robertson-Walker FRW) and continued expansion according to Hubble's Law. According to this description the dynamics of the universe would be determined by the amount of matter existing in it, which in turn determines the large-scale geometry of space-time. Recent observations of relic radiation in the Cosmic Microwave Background (CMB) have confirmed in essence the predictions of the Big Bang model and seem to corroborate the predictions of the FRW universe models with zero curvature (k = 0). More refined measures of CMB anisotropies ( Hinshaw 2009, Komatsu et al 2010) with the measurements of the supernovae (SNe Ia) at high redshift (Riess 1998, Perlmutter 1999) suggest the existence of a cosmic acceleration in accordance with the predictions of the model universe with constant cosmological (Λ≠0). This large-scale cosmic acceleration, also called dark energy, is today one of the most important enigmas of modern cosmology (see Peebles & Rastra 2003, Peebles 2020, Carroll & Tunner 1992; and references therein) . Even more disturbing is the apparent contradiction between the models of the universe with zero curvature (k = 0) in the FRW formalism and the total density parameter (Ω), according to which should be exactly equal to unity, but the observed density of matter is an order of magnitude lower than expected to play the null curvature (Overduin & Wesson 2008). Assuming that the dynamics of the universe is prescribed only by the force of gravity (as the only fundamental force in the astronomical scales) we encounter serious difficulties in describing the behavior of the Universe:-i-Can not explain the rotation curves of galaxies, which show its incompatibility with the virialized masses of the galaxies (Sofue 1999, Sanders 2002, Freese 2000).-ii-Into of the rich clusters of galaxies, the mass observed in the form of stars and the gas mass inferred from the X-ray diffuse emission is significantly less than that required to maintain these systems gravitationally stable (Overduin & Wesson 2008, Shirata et al 2005).-iii-In cosmological scales, the observed baryonic matter density is much lower than predicted by the FRW models with cosmological constant and curvature zero (Peebles & Rastra 2003, Peebles 2007, Carroll & Tunner 1992, Overduin & Wesson 2008) . The problem of missing mass appears to affect the dynamics at all length scale beyond the Solar System (Freese 2000). One solution has been to propose the requirement of missing material 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. In recent years there have been several alternatives to dark matter paradigm to explain the rotation curves of galaxies, or as alternatives to the TGR and the Big Bang cosmology. Among the first calls include MOND theories, which reproduced successfully the rotation curves of galaxies. But the formalism of the MOND theories (i) can not resolved the lack of dark matter at scales of clusters of galaxies (ii) neither the missing masses in the cosmological scale and (iii) is based on the modification of the Gravitation Universal Law to scales larger than the solar system, where the dependence of the force of gravity does not necessarily verify the law of the inverse of the square. In the same vein are the Theories of Moffat (Moffat 2006), who postulates a modification of the Universal Gravitation constant, although it is a very promising alternative to the paradigm of Dark Matter, faces problems that a variation of 25% or more in the constant acceleration of gravity would imply an abundance of helium incompatible with the observations (Reeves 1994, Melnikov 2009). Among the proposed changes to the TRG the most promising are those that postulate the existence of an additional scalar field metric tensor (Branks-Dicke theory) (Fujii & Maeda 2004) and within this so-called quintessence scenarios (Martin 2008) to postulate a new fundamental interaction, additional to gravity and electroweak interactions (electromagnetic and weak nuclear force) and strong nuclear force.While it is true that the validity of inverse square law of Newton's gravity is verified with precisions greater than 10^-8 for Eötvös-like experiments (Adelberger et al. 2003), there is no empirical evidence of their validity beyond the Solar System (Silverman 1987, Gundlach 2005); it is assumed true for estimating the mass of binary stars. For a review of the many theoretical speculations about deviations from the r^-2 law; see Adelberger and references therein (Adelberger et al. 2003). From the very beginning of cosmology and pre-relativistic ideas raised in the modification of gravity beyond the solar system (Goldman 1987 , Bondi 1951), but the experiments in laboratories on Earth and in the inner solar system, constitute a strong constraint for any alleged field the anti-gravity with range much greater than 1 AU (Goldman 1987). The universal character of Newton's Gravitation law was given by Kant in 1755, considering the deductive character of the planetary motion and the Leverrier’s prediction for the discovery of Neptune. Recall that galaxies acquire identity after the great debate Shapley-Curtis in 1920.Also Laplace and Seeliger theorized in the eighteenth century, modifications to the Law of gravitation (Bondi 1951; Seeliger 1895, Laves 1898). In relation to the hypothesis of non baryonic Dark Matter the history of science has shown many examples the paradigmatic assumptions to explain the behavior of the nature, which later were nonexistent and replaced by alternatives measurable. Such as the cycles and epicycles of Ptolemy, the ether before the advent of the Special Theory of Relativity, the “caloric” (as elementary substance) before the work of Joule and Carnot. In all cases, a review of the assumptions in the phenomenological description of the processes, led to a breakthrough in our understanding of natural reality.

1.2 Cosmic Acceleration: Dark Energy

We can propose for modification to the theory of gravitation Newton , but consistency is required to connect the new ideas with FRW models (Sanders 2002, Milgrom 2001) and the observables of the Big Bang model; such as the acoustic peaks in the CMB fluctuations, the density of matter, the age of the universe in terms of the Hubble constant, primordial nucleosynthesis (baryogenesis) and the formation of structures beginning the primordial fluctuations. On other hand, the gravitation of Newton postulates that the force of gravity has an infinite reach; even bigger than the radius of the observable Universe. Consequently implies that the rest-mass of the graviton is null, in contradiction with the Grand Unification Theories (GTU) and the detection of gravitational waves, which predict not null rest-mass (Chugreev 2017). Another biggest problem in the hot Big Bang cosmology, closely linked to the expansion of the Universe, is the evidence of the accelerated expansion, commonly referred as dark energy, whose understanding is still unfinished (Huterer & Shafer 2018; Debono & Smoot 2016; Genova-Santos 2020). Experiment to detect dark energy forces using atom interferometry (Sabulsky et al. 2019) shows no evidence of new forces those results had places stringent bounds on scalar field theories that modify General Relativity (GR) on large scales, as Chameleon and Symmetron theories of modified gravity. It is valid to ask if there are theoretical alternatives to Newton law of the gravity valid to large scale, concomitant with the astronomical observations and with the LFRW models; in the standard cosmology of the hot Big Bang. An alternative to solve both problems: dark energy and non-baryonic dark matter, are the theories of Modified Newtonian Dynamics (MoND) (Debono & Smoot 2016), such as non-local gravitation, establishing that the force of gravitation would be the result of two term generated by ordinary matter: a first term as Newton's law, and an additional term, representing the long-range contribution in the sense of the Mach (Falcon 2013, Falcon 2021).

Figure 1. Illustration of Mach`s principle: an Foucault pendu-lum period, in north pole, is exactly an sidereal day (Own source).

1.3 The March's principle

In the book “The Science of Mechanics”, Ernst Mach (Mach 1893) 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 tele-connection 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 (Einstein 1970) “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 (Bondi 1951), that is to say: the inertial field defined by the metric tensor (gmn) had to be completely determined by the distribution of masses and energies in the Universe (Tmn); 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 gmn from a given Tmn . 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. Similarly, any point of space in the adjacencies of the Local Group of galaxies, will be subjected to the gravitational interaction of the baryonic mass, corresponding to the galaxy clusters and the large-scale structures. As a result, there must be a global gravitational interaction that adds to the force that mediates between any galaxies pair. 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. Clearly, it is not possible to calculate explicitly that global contribution to the gravitational force between two particles. Einstein tried it, through the cosmological term Λ, but it remained pending how to model its equivalent in distance ranges stellar inside a particular galaxy, and to the interior of the galaxy clusters. But it is possible to propose alternatives for modeling the term of global inertia, heuristically constructed from the observations, to represent the gravitational contribution of the large-scale distribution of baryonic matter in the local universe

1.4 Purpose and goal

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 non zero 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 Falcon 2013, Falcon & Aguirre 2014, 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 .For this, in section 1 the basic ideas of the model are raised; starting with the phenomenology and physical arguments. The cosmological consequences of the inclusion of a large-scale term in the Newtonian law of gravitation are analyzed through an ᴧFRW-Cosmology model (section 2), then they are exposed several astrophysical implications (section 3) such as the Virial Theorem and the solve the Zwicky's paradox, the large-scale variation of Kepler's third Law in globular clusters, the gravitational redshift, the Jean's mass together a preliminary study of its implications in the Baryonic Acoustic Oscillations (BAO) and the Cosmic Microwave Background (CMB), and Pioneer anomalies. Finally the conclusions are shown in the last section.

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