Quantum Discrete Levels of the Universe from the Early Trans-Planckian Vacuum to the Late Dark Energy

Abstract

The standard model of the universe is further completed back in time before inflation in agreement with observations, classical-quantum gravity duality and quantum space-time. Quantum vacuum energy bends the space-time and produces a constant curvature de Sitter background. We link de Sitter universe and the cosmological constant to the (classical and quantum) harmonic oscillator. Quantum discrete cosmological levels are found: size, time, vacuum energy, Hubble constant and gravitational (Gibbons-Hawking) entropy from the very early trans-planckian vacuum to the classical today vacuum energy. For each level $n=0,1,2,...$, the two: post and pre (trans)-planckian phases are covered: In the post-planckian universe, the levels (in planck units) are: Hubble constant $H_n=1/\sqrt{(2n+1)}$, vacuum energy ${\Lambda }_n=1/(2n+1)$, entropy $S_n=(2n+1)$. As $n$ increases, radius, mass and $S_n$ increase, $H_n$ and ${\Lambda }_n$ decrease and {\it consistently} the universe {\it classicalizes}. In the pre-planckian (trans-planckian) phase, the quantum levels are: $H_{Qn}=\sqrt{(2n+1)},{\Lambda }_{Qn}=(2n+1)/1,S_{Qn}=1/(2n+1)$, $Q$ denoting quantum. The $n$-levels cover {\it all} scales from the far past highest excited trans-planckian level $n=10^{122}$ with finite curvature, ${\Lambda }_Q=10^{122}$ and minimum entropy $S_Q=10^{-122}$, $n$ decreases till the planck level $(n=0)$ and enters the post-planckian phase e.g: $n=1,2,...,n_{inflation}=10^{12},...,n_{cmb}=10^{114},...,n_{reoin}=10^{118},...,n_{today}=10^{122}$ with the most classical value $H_{today}=10^{-61}$, ${\Lambda }_{today}=10^{-122}$, $S_{today}=10^{122}$. We implement the Snyder-Yang algebra in this context yielding a consistent group-theory realization of quantum discrete de Sitter space-time, classical-quantum gravity duality symmetry and a clarifying unifying picture.(Abridged)

arXiv commentPublished version in Phys Rev D, 54 pages, 1 figure, new material. arXiv admin note: text overlap with arXiv:1912.06655

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