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

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 $H_n=1\mathrm{/}\sqrt{(2n+1)}$, vacuum energy ${\mathrm{\Lambda }}_n=1\mathrm{/}(2n+1)$, entropy $S_n=(2n+1)$. As $n$ increases, radius, mass and $S_n$ increase, $H_n$ and ${\mathrm{\Lambda }}_n$ decrease and {\it consistently} the universe {\it classicalizes}. In the pre-planckian (trans-planckian) phase, the quantum levels are: $H_{Qn} = \sqrt{(2n + 1)},; \Lambda_{Qn} = (2n + 1)/1,; S_{Qn} = 1/(2n + 1)$,$Q$denotingquantum\mathrm{.}The$n$-levels cover {\it all} scales from the far past highest excited trans-planckian level $n={10}^{122}$ with finite curvature, ${\mathrm{\Lambda }}_Q={10}^{122}$ 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 $H_{today}={10}^{-61}$, ${\mathrm{\Lambda }}_{today}={10}^{-122}$, $S_{today}={10}^{122}$. 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)