Quantum discrete levels of the Universe from the early trans-Planckian vacuum to the late dark energy

We go forward in completing the Standard Model of the Universe back in time with Planckian and trans-Planckian physics before inflation in agreement with observations, classical-quantum gravity duality, and quantum space-time. The quantum vacuum energy bends the space-time and produces a constant curvature de Sitter background. We link the de Sitter Universe and the cosmological constant to the (classical and quantum) harmonic oscillator. We find the quantum discrete cosmological levels: size, time, vacuum energy, Hubble constant, and gravitational (Gibbons-Hawking) entropy and temperature from the very early trans-Planckian vacuum to the classical vacuum energy today. For each level n =0 ,1 ,2 ,… , the two post- and pre-(trans)-Planckian phases are covered: In the post-Planckian Universe t_planck≡t_P≤t ≤10⁶¹t_P, the levels (in Planck units) are Hubble constant H_n=1 /√{(2 n +1 ) }, vacuum energy Λ_n=1 /(2 n +1 ), and entropy S_n=(2 n +1 ). As n increases, radius, mass, and S_n increase, H_n and Λ_n decrease, and consistently the Universe classicalizes. In the pre-Planckian (trans-Planckian) phase 10^-61t_P≤t ≤t_P, the quantum levels are H_{Q n}=√{(2 n +1 ) }, Λ_{Q n}=(2 n +1 ), and S_{Q n}=1 /(2 n +1 ), Q denoting quantum. The n levels cover all scales from the far past highest excited trans-Planckian level n =10¹²² with finite curvature, Λ_Q=10¹²², and minimum entropy S_Q=10^-122; n decreases till the Planck level (n =0 ) with H_planck=1 =Λ_planck=S_planck and enters the post-Planckian phase, e.g., n =1 ,2 ,…, n_inflation=10¹²,… , n_cmb=10¹¹⁴,… , n_reoin=10¹¹⁸,…, and n_today=10¹²², with the most classical value H_today=10^-61, Λ_today=10^-122, and S_today=10¹²². 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.