A mass scale law connecting cosmophysics to microphysics

Abstract

We introduce different mass scales corresponding to the Universe, fermion stars, mini boson stars, Planck black holes, the electron, the neutrino, and the cosmon. These mass scales are obtained by combining the maximum mass of fermion stars, boson stars and soliton stars set by quantum mechanics and general relativity with the Eddington relation connecting the mass me of the electron to the cosmological constant Λ. In this manner, we can express the mass of these objects in terms of the fundamental constants of physics G, c, ħ and Λ. By normalizing the mass Ma of these objects by the Planck mass MP, we find that Ma/MP∼χa/6, where χ∼ρP/ρΛ∼10120 is the “largest large number” in Nature (the ratio of the Planck density on the Einstein cosmological density) and a=3,2,1,0,−1,−2,−3 for the Universe, fermion stars, mini boson stars, Planck black holes, the electron, the neutrino, and the cosmon respectively. This formula suggests an interesting symmetrical mass scale law connecting cosmophysics (Universe, fermion stars, mini boson stars) to microphysics (electron, neutrino, cosmon). A generalization of this law including the earth mass and another neutrino mass scale is proposed. We highlight an accurate form of Eddington relation me≃α(Λħ4/G2)1/6 or Λ≃G2me6/α6ħ4=α−6(me/MP)6lP−2, where α=e2/ħc≃1/137 is the fine structure constant, and provide different heuristic derivations of this relation. We consider another relation Λ=Gc(mΛ∗)4/ħ3=(mΛ∗/MP)4lP−2, where mΛ∗=(Λħ3/Gc)1/4 is the neutrino mass. We relate these relations to the two expressions of the vacuum energy density given by Zeldovich in 1968. We suggest that the dark energy particle (cosmon) of mass mΛ=ħΛ/c could be a quantum of mass and entropy so that Λ=mΛ2c2/ħ2=(mΛ/MP)2lP−2. We relate the large number 10120 to the total number of degrees of freedom (or number of cosmons) in the universe. Finally, we give hints how to possibly solve the cosmological constant problem, the cosmic coincidence problem, and the large number coincidence problem.