A Geometric Framework for the Universe Energy Spectrum

Abstract: In this article we present a framework for the energy spectrum of the Universe by regarding the energy spectrum of the Universe just like an atom, i.e. $E(q)=M_pe^{-q\xi}$, with discrete energy levels, from the Planck scale to the lightest neutrino mass. To establish this spectrum analytically in a heuristic way and starting from two known constants, $G_N$ and $G_F$, we establish a Hierarchy Equation, $\xi = W\left(\sqrt{2}M_p/v\right)$, to bridge them through the Lambert W function to the dimensionless energy geometric structure constant $\xi = 35.227$, which is the geometric fundamental unit for all energy scales. Concerning the detailed energy spectrum within the electroweak vacuum, $v$, we propose a Mass Generation Equation (MGE), $[G][m]=[v]$, and the masses of all the fundamental particles can be obtained analytically. We have proposed also some analytical formulae for the cosmology, CKM and PMNS related quantities, such as the mixing, oscillation angles, and CP phases. By investigating the energy geometrical structure in $q$ space, we find that there exist inherent fractal and holographic structures within the whole spectrum based on the  UV/IR duality connections, such as $v^2=m_1M_s$, with the lightest neutrino mass $m_1=1.24\times 10^{-13}\text{GeV}$ and the string energy scale $M_s=\sqrt{2}M_p/\xi=4.898\times 10^{17}\text{GeV}$. In this article, we establish many energy scale formulae, such as $M_s$, $M_{mono}$, $T_{reh}$, $M_{GUT}$, $H_{inf}$, $f_a$, $M_{L}$, $\Lambda_{SUSY}$, $\Lambda_{QCD}$, $\Gamma_{vac}$, $\Lambda$ (cosmology constant), $m_{vac}$, and particle mass predictions, such as $m_1$, $m_2$, $m_3$, $M_{dark}$, $m_a$ and $m_{\beta\beta}$, which could be measured by the corresponding experiments. These formulae are offered as conjectures to stimulate further theoretical and experimental scrutiny.