Derivation of three fundamental masses and large numbers hypothesis by dimensional analysis

Abstract

Three mass dimension quantities have been derived by dimensional analysis by means of fundamental constants – the speed of light in vacuum (c), the gravitational constant (G), the Planck constant (h_bar) and the Hubble constant (H). The extremely small mass m1 ~ (h_bar*H)/c^2 ~ 10^(-33) eV has been identified with the Hubble mass mH, which seems close to the graviton mass mG. The enormous mass m2 ~ c^3/(G*H) ~ 10^53 kg is close to the mass of the Hubble sphere and practically coincides with the Hoyle-Carvalho formula for the mass of the observable universe. The third mass m3 ~ [(H*h_bar^3)/G^2]^(1/5) ~ 10^7 GeV could not be unambiguously identified at present time. Besides, it has been found remarkable fact that the Planck mass mPl ~ Sqrt [(h_bar*c)/G] appears geometric mean of the extreme masses m1 and m2. Finally, the substantial large number N = Sqrt [c^5/(2*G*h_bar*H^2) ≈ 5.73×10^60 has been derived relating cosmological parameters (mass, density, age and size of the observable universe) and fundamental microscopic properties of the matter (Planck units and Hubble mass). Thus, a precise formulation and proof of Large Numbers Hypothesis (LNH) has been found.