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

International audienceWe 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 tplanck≡tP≤t≤1061tP, the levels (in Planck units) are Hubble constant Hn=1/(2n+1), vacuum energy Λn=1/(2n+1), and entropy Sn=(2n+1). As n increases, radius, mass, and Sn increase, Hn and Λn decrease, and consistently the Universe classicalizes. In the pre-Planckian (trans-Planckian) phase 10-61tP≤t≤tP, the quantum levels are HQn=(2n+1), ΛQn=(2n+1), and SQn=1/(2n+1), Q denoting quantum. The n levels cover all scales from the far past highest excited trans-Planckian level n=10122 with finite curvature, ΛQ=10122, and minimum entropy SQ=10-122; n decreases till the Planck level (n=0) with Hplanck=1=Λplanck=Splanck and enters the post-Planckian phase, e.g., n=1,2,…, ninflation=1012,…, ncmb=10114,…, nreoin=10118,…, and ntoday=10122, with the most classical value Htoday=10-61, Λtoday=10-122, and Stoday=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