Universe as a representation of affine and conformal symmetries

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type Ia supernova Iuminosity distances - redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of 'squeezed oscillator'. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators