EXTENDED MASS RELATION FOR SEVEN FUNDAMENTAL MASSES AND NEW EVIDENCE OF LARGE NUMBERS HYPOTHESIS

A previously derived mass relation has been extended to seven equidistant fundamental masses covering an extremely large mass range from ~ 10 kg to ~ 10 kg. Six of these masses are successfully identified as mass of the observable universe, Eddington mass limit of the most massive stars, mass of hypothetical quantum “Gravity Atom” whose gravitational potential is equal to electrostatic potential e/S, Planck mass, Hubble mass and mass dimension constant relating masses of stable particles with coupling constants of fundamental interactions. The seventh mass, ~ 10 kg remains unidentified and could be considered as a prediction of the suggested mass relation for an unknown fundamental mass, potentially a yet unobserved light particle. First triad of these masses describes macro objects, the other three masses belong to particle physics masses, and the Planck mass appears intermediate in relation to these two groups. Additionally, new evidences of Dirac Large Numbers Hypothesis (LNH) have been found in the form of series of ratios relating cosmological parameters and quantum properties of space-time. A very large number on the order of 5×10 connects mass, density, age and size of the observable universe with Planck mass, density, time and length, respectively.


INTRODUCTION
Discovery of theoretical or empirical mass relations for the many various particles is a great challenge for the recent high-energy physics and astrophysics, and derivation of mass relations covering a very large range of particle masses is most desirable. Known are a few formulas connecting the masses of particles having similar properties, one such, is Hadron's multiplets (octets and decuplets of particles having close masses).
Though imprecise, one of the first attempts to empirically derive 'Balmer's law' for several particles has been attempted from Nambu (1952), wherein, e n nm m 137 is the mass of the nth particle, m e is mass of the electron, and n is an integer or half-odd. Based on SU(3) symmetry, the Gell-Mann -Okubo mass formula (Gell-Mann, 1961;Okubo, 1962) has been derived for baryon decuplet: , where m ∆ , m Σ , m Ξ and m Ω are the masses of respective hyperons. This formula successfully predicted the mass for the then undiscovered Ω − hyperon. The mass relations of Georgi-Jarlskog (1979) ensue from the SO(10) model and relate masses of charged leptons (e, µ and τ) and down-type quark (d, s and b) 3 / and m τ = m b . However, these mass relations yield results that deviate significantly as compared to experimental data. It is postulated in (Barut, 1979) that a quantized magnetic self-energy of magnitude ) 2 /( 3 4 α n m e be added to the rest mass of a lepton to get the next heavy lepton in the chain e, µ, τ,⋯, with n = 1 for µ, n = 2 for τ, etc. Here, α is the fine structure constant, m e ≈ 0.511 MeV is the mass of the electron and n is a new quantum number. Thus it was predicted M τ =1786.08 MeV, and for the next lepton M δ =10293.7 MeV. Koide (1993)  neutrino ν e and graviton) with coupling constants ) 0 ( i α of the four interactions, and i = 1, 2, 3, 4. This mass relation covers an extremely wide range of values, exceeding 40 orders of magnitude and predicts a graviton mass on the order of 10 -69 kg. Found in (Forsythe, 2009) is the derived mass relation: (1), where N ~ 6.02×10 23 is a large pure number and n = 1,2,3,4.
This mass formula produces four equidistant masses covering a large range of 61 orders of magnitude. Mass M 1 ~ 2.18×10 -8 kg is apparent Planck mass 80×10 12 kg is the apparent mass of a hypothetical quantum "Gravity Atom" whose gravitational potential is equal to electrostatic potential S e / 2 , M 3 ~ 6.62×10 32 kg has not been identified and M 4 ~ 1.16×10 53 kg is the assumed proper mass of the observable universe. Now, in the present paper, we extend mass relation (1) to produce seven equidistant fundamental masses covering extremely large mass range of 122 powers of magnitude. Dirac (1937) noticed that the ratio of the age of the universe H -1 , the inverse of the Hubble parameter, and the atomic unit of time, in a hydrogen atom is 2.27×10 39 , were G is the Newtonian constant of gravitation and m e and m p are electron and proton masses respectively. These "coincidences" hint at a possible connection between macro and microphysical world known as Dirac Large Numbers Hypothesis (LNH). Many other interesting ratios have been found approximately relating some astrophysical (cosmological) parameters and microscopic properties of the matter. For example Jordan (1947) noted that the mass ratio for a typical star and an electron is of the order of 10 60 . Narlikar (1977) shows that the ratio of the observable universe radius, cH -1 , and the classical electron radius, 2 2 / c m e e is exactly equal to N D . Additionally, the ratio of the electron mass and Hubble mass parameter 2 / c H h is 3.39×10 38 (Cetto et al., 1986). Here ) 2 is the reduced Planck constant and H is the Hubble constant. Peacock (1999) points out that the ratio of Hubble distance cH -1 and Planck length l P is on the order of 10 60 . Besides, the ratio of Planck density ρ P and recent critical density of the universe ρ c is found to be on the order of 10 121 (Andreev and Komberg, 2000). Further, the ratio of observable universe mass and Planck mass is on the order of 10 61 (Shemizadeh, 2002). These ratios between astrophysical parameters and microscopic properties of matter result mostly in large numbers that roughly agree with order of magnitude accuracy. Valev (2012) derived a series of ratios relating cosmological parameters (mass M, density ρ c , age H -1 and size cH -1 of the observable universe) and Planck (mass m P , density ρ P , time t P and length l P ) respectively, resulting in a very large number N V , wherein m P is defined as the mass whose reduced Compton wavelength and Schwarzschild radius r s are equal, l P is identical with r s , and ρ P is defined as the density of a sphere having mass m P and radius l P .
These ratios exactly connect cosmological and quantum parameters of space-time and appear to be a precise formulation and proof of LHN. In this paper, we have found new evidences in support of LNH connecting cosmological parameters and microscopic properties of matter.

Review of Mass Relation Concerning Four Fundamental Masses
In the previous paper (Forsythe, 2009), Newton's law of universal gravitation is derived, based on postulated mass/energy resonance waves, wherein the apparent Newtonian constant of gravitation factors as: where m e is electron rest mass, φ λ the resonance wavelength, φ m the associated particle mass, and N is a large pure number, curiously comparably with N A , the 2006 recommended numerical value of Avogadro's number, and in terms of the fine structure constant α, and π, is shown to be given by: The Planck mass by convention is Planck, 1906). Therefore, it follows from Eq. (3) that the apparent Planck mass is given by: Additionally shown is that the resonance wavelength is equal to twice the first Boar orbit thus leading directly to: It is known that the fine structure constant, the coupling constant of electromagnetic interaction, i. e. a measure of its strength, is determined by the formula ) /( 2 c e h = α . Taking into consideration this formula, we find from Eq. (3) that: In Section II of paper (Forsythe, 2009), a hypothetical quantum "Gravity Atom" has been proposed, comprised of an electrically neutral central mass M G orbited by an electrically neutral particle having electron mass m e such that the gravitational potential S m GM e G / is equal to an electrostatic potential S e / 2 and S, the orbital radius, is a Bohr orbit. Thus, , that in conjunction with Eq. (7) results in: It is also of interest to note that this is the mass for which the Schwarzschild radius is equal to twice the classical electron radius.
Noted in (Forsythe, 2009) is that examination of Eqs. (5) and (8)  (1), that in conjunction with Eq. (6) can also be expressed as where n is the placement within the series. Employing Eq.
(1) and beginning at n = 1, it is found that: Identified above is the physical significance attributed to masses M 1 and M 2 . Mass M 4 appears to be well within the range of estimates for the observable universe proper mass M u (Carvalho, 1995;Valev, 2014) and as such, it represents the upper limit of the series.

Extended Mass Relation for Seven Fundamental Masses, a New Fundamental Constant K and the Hubble Parameter
Upon extending the series downwards to 0 ≤ n , we obtain: The current best estimates of H 0 center around about 70 km s -1 Mps -1 . Thus, when m H , the Hubble mass (Maor and Brustein, 2003;Gazeau and Toppan, 2010), is defined as:  (20) and (21) must then be multiplied by 2π to preserve the equalities, and Eq. (22) is still the final result.
As was proposed in (Forsythe, 2012), predicated upon the rate of cosmic expansion apparently transitioning from deceleration to acceleration at redshift z ~ 0.5 (Perlmutter et al., 1999), the deceleration parameter must have passed through a zero null point at transition, as the opposing operatives of cosmic expansion reached a transient state of equilibrium. Intuitively it would seem that the Hubble parameter at that juncture H eq , the tipping point between deceleration and acceleration, must be tied to the mass of the universe via means of a unique relationship that existed at that juncture, as developed through Eqs. (19), (20), (21), and (22), leading to Eq. (23). However, it does not necessarily follow that the Hubble parameter is increasing along with the accelerating rate of cosmic expansion. Some theoretical considerations suggest that the Hubble parameter has now assumed a truly constant value in time and space. Others predict that even as the expansion accelerates, the Hubble parameter will continue to decrease asymptotically, approaching a limiting value of about 62 km s -1 Mpc -1 , as the influence of the cosmological constant becomes more and more dominant over the contribution of matter after several billions of years and a several fold increase in the scale factor. It is thus reasonable to propose that H 0 , the present day Hubble parameter, and H eq are essentially identical. Thus: A theoretical value for H 0 of 68.66±0.1 km s -1 Mps -1 , obtained via an entirely independent approach (Bukalov, 2002), is in excellent agreement with the above. Since by convention, the square of the Planck mass is ) 2 /( G hc π Eq. (23) can be restated as:

Review of Three Fundamental Masses Obtained by Dimensional Analysis
In previous paper (Valev, 2013), three fundamental masses have been derived by dimensional analysis, namely: In form, Eq. (26) coincides closely with Eq. (29) and the two are an identity when the dimensionless parameter k, on the order of unity, is identical with 2/π.
The papers (Forsythe, 2009;Forsythe, 2012) do not attribute any physical significance to mass M 3 ~ 6.63×10 32 kg in the original n 1 through n 4 series. Recently we have identified this mass with the Eddington stellar mass limit where the outward pressure of the star's radiation balances the inward gravitational force (Vink et al., 2011;Crowther et al., 2011). Additionally, we have identified the mass M 0 ~ 1.25×10 -28 kg as exactly coinciding with the mass dimension constant in a basic mass equation from paper (Valev, 2008) relating masses of stable particles and coupling constants of the four fundamental interactions. It is interesting that this mass is approximately a halfcharged pion mass . Mass M (-1) ~ 7.15×10 -49 kg is presently unidentified and could feasibly be regarded as a prediction by the suggested model, Eq. (9), for a fundamental, albeit as yet unobserved light particle. Finally, mass M (-2) ~ 4.10×10 -69 kg in the extended series is easily identifiable with the Hubble mass Eq. (19) as 0.5πm H . It is of further interest to note that the extended mass series includes seven equidistant fundamental masses covering a mass interval of 122 orders of magnitude, and that masses M (-2) , M (-1) and M 0 are particle physics masses, whereas the masses M 2 , M 3 and M 4 describe macro objects, and the Planck mass M 1 appears intermediate in relation to these two groups. In fact, it is easily shown that the Planck mass, as given by Eq. (9), is the geometric mean of the extreme masses M (-2) and M 4 as given by Eqs. (15) and (12), as is the geometric mean of masses m 1 and m 2 from Eqs. (28) and (29) when k =1. Valev mass m 3 from Eq. (30) has not yet been identified and could be regarded as a prediction for unknown fundamental mass, most likely a yet unobserved very heavy particle.

NEW EVIDENCES OF DIRAC LARGE NUMBERS HYPOTHESIS
Recalling Eq. (2) and the definition of terms therein, it is found that N V ~ 5.73×10 60 when the defined terms are evaluated according to: is recent density of the universe equal to the critical one; H -1 , the age of the universe and cH -1 is the Hubble distance.
The Eq.
(2) ratios appear very important because they relate cosmological parameters and the fundamental microscopic properties of matter. The Planck units imply quantization of space-time at extremely short range. Thus, the ratios represent connection between cosmological and quantum parameters of space-time and thus appear to be a precise formulation and proof of LHN. In addition, the very large number N V and Dirac large number N D (Dirac, 1937) seem connected by the approximate formula: We now construct a similar series to (2) involving ratios of the same parameters producing the very large, number N VF , as follows: is apparent proper mass of the universe; and G is according to Eq. (3). These ratios also represent a connection between cosmological and quantum parameters of space-time and so likewise appear to be possible new evidences of LNH. Recalling Eq. (17) Thus, by independent approaches it is apparent that we obtain very similar results, (31), (33) and (2)

CONCLUSIONS
Mass relation (1) obtained in (Forsythe, 2009) has been extended from n = -2 to n = 4. The result is seven equidistant fundamental masses M n , covering a mass interval of 122 orders of magnitude, have been obtained. Six of these masses are successfully identified, namely M 1 ~ 2.18×10 -8 kg the apparent Planck mass , that is very important in resent particle physics. The mass M 2 ~ 3.80×10 12 kg is the central mass of a hypothetical quantum "Gravity Atom" whose gravitational potential S m GM e G / is equal to electrostatic potential S e / 2 and S is a Bohr orbit radius and the mass M 3 ~ 6.63×10 32 kg is of the order of the Eddington mass limit of the most massive stars. The mass M 4 ~ 1.16×10 53 kg is close to the mass of the Hubble sphere and most probably appears to be mass of the observable universe. The mass M 0 ~ 1.25×10 -28 kg coincides with a mass dimension constant in a basic mass equation relating masses of stable particles and coupling constants of the four interactions; approximately a half charged pion mass. The mass M (-2) ~ 4.10×10 -69 kg is easily identifiable with the Hubble mass as 0.5πm H . The mass M (-1) ~ 7.15×10 -49 kg remains yet unidentified and could be regarded as a prediction by the suggested mass relation for unknown fundamental mass, most likely a yet unobserved light particle. Apparently, masses M (-2) , M (-1) and M 0 are particle physics masses, whereas the masses M 2 , M 3 and M 4 describe macro objects, and the Planck mass M 1 appears intermediate in relation to these two groups.
Finally, new evidences of LNH have been found in the form of series of ratios relating cosmological parameters and quantum properties of space-time. In addition, the very large number h α π 5.31×10 60 connects mass, density, age and size of the observable universe with Planck mass, density, time and length respectively, and K is apparently a unique and new fundamental constant.

ACKNOWLEDGMENT
We would like to thank James R. Johnson and Andrew W. Beckwith for encouraging discussions.