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 Hn = 1/(2n + 1), vacuum energy n = 1/(2n + 1), entropy Sn = (2n + 1). As n increases, radius, mass and Sn increase, Hn and n decrease and consistently the universe classicalizes. In the pre-planckian (trans-planckian) phase, the quantum levels are: HQn = (2n + 1),\; Qn = (2n + 1)/1,\; 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, Q = 10122 and minimum entropy SQ = 10-122, n decreases till the planck level (n = 0) and enters the post-planckian phase e.g: n = 1, 2,...,ninflation = 1012,... ,ncmb = 10114,...,nreoin = 10118,...,ntoday = 10122 with the most classical value Htoday = 10-61, today = 10-122, 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.(Abridged)